#set document(title: "7.1 Multiply and Divide Rational Expressions", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 7.1#h(0.6em)Multiply and Divide Rational Expressions #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Simplify: #math.equation(block: false, alt: "the fraction 90 y over 15 y squared .")[$frac(90 y, 15 y^(2)) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "the fraction 6 over y")[$frac(6, y)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Multiply: #math.equation(block: false, alt: "the fraction 14 over 15 times the fraction 6 over 35 .")[$frac(14, 15) · frac(6, 35) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 25")[$frac(4, 25)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Divide: #math.equation(block: false, alt: "the fraction 12 over 10 divided by the fraction 8 over 25 .")[$frac(12, 10) ÷ frac(8, 25) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "the fraction 15 over 4")[$frac(15, 4)$] ] ] We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a #strong[rational expression]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Rational Expression] A rational expression is an expression of the form #math.equation(block: false, alt: "the fraction p over q ,")[$frac(p, q) ,$] where #emph[p] and #emph[q] are polynomials and #math.equation(block: false, alt: "q not equal to 0 .")[$q ≠ 0 .$] ] Here are some examples of rational expressions: #math.equation(block: true, alt: "minus the fraction 24 over 56, the fraction 5 x over 12 y, the fraction 4 x plus 1 over x squared minus 9, the fraction 4 x squared plus 3 x minus 1 over 2 x minus 8")[$− frac(24, 56) & & & #h(3em) frac(5 x, 12 y) & & & #h(3em) frac(4 x + 1, x^(2) − 9) & & & #h(3em) frac(4 x^(2) + 3 x − 1, 2 x − 8)$]Notice that the first rational expression listed above, #math.equation(block: false, alt: "minus the fraction 24 over 56")[$− frac(24, 56)$], is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero. We will do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications. === Determine the Values for Which a Rational Expression is Undefined If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator. When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero. So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Determine the values for which a rational expression is undefined.] + Set the denominator equal to zero. + Solve the equation. ] #examplebox("Example 1")[][ Determine the value for which each rational expression is undefined: ⓐ #math.equation(block: false, alt: "the fraction 8 a squared b over 3 c")[$frac(8 a^(2) b, 3 c)$] ⓑ #math.equation(block: false, alt: "the fraction 4 b minus 3 over 2 b plus 5")[$frac(4 b − 3, 2 b + 5)$] ⓒ #math.equation(block: false, alt: "the fraction x plus 4 over x squared plus 5 x plus 6 .")[$frac(x + 4, x^(2) + 5 x + 6) .$] #solutionbox[ The expression will be undefined when the denominator is zero. ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 8 a squared b over 3 c")[$#h(5em) frac(8 a^(2) b, 3 c)$]]), [Set the denominator equal to zero and solve #linebreak() for the variable.], [#math.equation(block: false, alt: "3 c equals 0")[$#h(5em) 3 c = 0$]], [], [#math.equation(block: false, alt: "c equals 0")[$#h(5em) c = 0$]], [], [#math.equation(block: false, alt: "the fraction 8 a squared b over 3 c is undefined for c equals 0 .")[$#h(5em) frac(8 a^(2) b, 3 c) #h(0.2em) "is undefined for" #h(0.2em) c = 0 .$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 4 b minus 3 over 2 b plus 5")[$#h(5em) frac(4 b − 3, 2 b + 5)$]]), [Set the denominator equal to zero and solve #linebreak() for the variable.], [#math.equation(block: false, alt: "2 b plus 5, equals, 0; 2 b, equals, −5; b, equals, minus the fraction 5 over 2")[$#h(5em) 2 b + 5 & = & 0 \ 2 b & = & −5 \ b & = & − frac(5, 2)$]], [], [#math.equation(block: false, alt: "the fraction 4 b minus 3 over 2 b plus 5 is undefined for b equals minus the fraction 5 over 2 .")[$#h(5em) frac(4 b − 3, 2 b + 5) #h(0.2em) "is undefined for" #h(0.2em) b = − frac(5, 2) .$]], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction x plus 4 over x squared plus 5 x plus 6")[$frac(x + 4, x^(2) + 5 x + 6) #h(1.4em)$]]), [Set the denominator equal to zero and solve #linebreak() for the variable.], [#math.equation(block: false, alt: "x squared plus 5 x plus 6 equals 0 ; open parenthesis x plus 2 close parenthesis open parenthesis x plus 3 close parenthesis equals 0 ; x plus 2 equals 0 or x plus 3 equals 0 ; x equals −2 or x equals −3")[$x^(2) + 5 x + 6 = 0 #h(1em) \ ( x + 2 ) ( x + 3 ) = 0 #h(1em) \ x + 2 = 0 #h(0.5em) "or" #h(0.5em) x + 3 = 0 #h(1em) \ x = −2 #h(0.2em) "or" #h(0.2em) x = −3 #h(0.2em)$]], [], [#math.equation(block: false, alt: "the fraction x plus 4 over x squared plus 5 x plus 6 is undefined for x equals −2 or x equals −3 .")[$frac(x + 4, x^(2) + 5 x + 6) #h(0.2em) "is undefined for" #h(0.2em) x = −2 #h(0.2em) "or" #h(0.2em) x = −3 .$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine the value for which each rational expression is undefined. ⓐ #math.equation(block: false, alt: "the fraction 3 y squared over 8 x")[$frac(3 y^(2), 8 x)$] ⓑ #math.equation(block: false, alt: "the fraction 8 n minus 5 over 3 n plus 1")[$frac(8 n − 5, 3 n + 1)$] ⓒ #math.equation(block: false, alt: "the fraction a plus 10 over a squared plus 4 a plus 3")[$frac(a + 10, a^(2) + 4 a + 3)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals 0")[$x = 0$] ⓑ #math.equation(block: false, alt: "n equals minus the fraction 1 over 3")[$n = − frac(1, 3)$] #linebreak() ⓒ #math.equation(block: false, alt: "a equals −1 , a equals −3")[$a = −1 , a = −3$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine the value for which each rational expression is undefined. ⓐ #math.equation(block: false, alt: "the fraction 4 p over 5 q")[$frac(4 p, 5 q)$] ⓑ #math.equation(block: false, alt: "the fraction y minus 1 over 3 y plus 2")[$frac(y − 1, 3 y + 2)$] ⓒ #math.equation(block: false, alt: "the fraction m minus 5 over m squared plus m minus 6")[$frac(m − 5, m^(2) + m − 6)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "q equals 0")[$q = 0$] ⓑ #math.equation(block: false, alt: "y equals minus the fraction 2 over 3")[$y = − frac(2, 3)$] #linebreak() ⓒ #math.equation(block: false, alt: "m equals 2 , m equals −3")[$m = 2 , m = −3$] ] ] === Simplify Rational Expressions A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a #strong[simplified rational expression] has no common factors, other than 1, in its numerator and denominator. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Simplified Rational Expression] A rational expression is considered simplified if there are no common factors in its numerator and denominator. ] For example, #math.equation(block: true, alt: "the fraction x plus 2 over x plus 3 is simplified because there are no common factors of x plus 2 and x plus 3 .; the fraction 2 x over 3 x is not simplified because x is a common factor of 2 x and 3 x .")[$frac(x + 2, x + 3) #h(0.2em) "is simplified because there are no common factors of" #h(0.2em) x + 2 #h(0.2em) "and" #h(0.2em) x + 3 . \ frac(2 x, 3 x) #h(0.2em) "is not simplified because" #h(0.2em) x #h(0.2em) "is a common factor of" #h(0.2em) 2 x #h(0.2em) "and" #h(0.2em) 3 x .$]We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Equivalent Fractions Property] If #emph[a], #emph[b], and #emph[c] are numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 ,")[$b ≠ 0 , c ≠ 0 ,$] #math.equation(block: true, alt: "then the fraction a over b equals the fraction a times c over b times c and the fraction a times c over b times c equals the fraction a over b .")[$"then" #h(0.5em) frac(a, b) = frac(a · c, b · c) #h(1em) "and" #h(1em) frac(a · c, b · c) = frac(a, b) .$] ] Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0")[$b ≠ 0 , c ≠ 0$] clearly stated. To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property. Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum. #figure(figph[The rational expression is the quantity 2 times 3 times 7 divided by the quantity 3 times 5 times 7 are 3 and 7. Its common factors are 3 and 7, which are factors of the product. When they are removed, the result is two-fifths. The rational expression is the product of 3 x and the quantity x minus 9 divided by the product of 5 and the quantity x minus 9. The common factor is x minus 9, which is a factor of the product. When it is removed, the result is 3 x divided by 5. The rational expression is the quantity x plus 5 divided by 5. There is an x both the numerator and denomiantor. However, it is a term of the sum in the numerator. The rational expression has no common factors.], alt: "The rational expression is the quantity 2 times 3 times 7 divided by the quantity 3 times 5 times 7 are 3 and 7. Its common factors are 3 and 7, which are factors of the product. When they are removed, the result is two-fifths. The rational expression is the product of 3 x and the quantity x minus 9 divided by the product of 5 and the quantity x minus 9. The common factor is x minus 9, which is a factor of the product. When it is removed, the result is 3 x divided by 5. The rational expression is the quantity x plus 5 divided by 5. There is an x both the numerator and denomiantor. However, it is a term of the sum in the numerator. The rational expression has no common factors.", caption: none) Removing the #emph[x]’s from #math.equation(block: false, alt: "the fraction x plus 5 over x")[$frac(x + 5, x)$] would be like cancelling the 2’s in the fraction #math.equation(block: false, alt: "the fraction 2 plus 5 over 2 !")[$frac(2 + 5, 2) !$] #examplebox("Example 2")[How to Simplify a Rational Expression][ Simplify: #math.equation(block: false, alt: "the fraction x squared plus 5 x plus 6 over x squared plus 8 x plus 12")[$frac(x^(2) + 5 x + 6, x^(2) + 8 x + 12)$]. #solutionbox[ #figure(figph[Step 1 is to factor the numerator and denominator completely in the rational expression, the quantity x squared plus 5 x plus six divided by the quantity x squared 8 x plus 12. The numerator, x squared plus 5 x plus six, factors into the quantity x plus 2 times the quantity x plus 3. The denominator, x squared 8 x plus 12, factors into the quantity x plus 2 times the quantity x plus 6.], alt: "Step 1 is to factor the numerator and denominator completely in the rational expression, the quantity x squared plus 5 x plus six divided by the quantity x squared 8 x plus 12. The numerator, x squared plus 5 x plus six, factors into the quantity x plus 2 times the quantity x plus 3. The denominator, x squared 8 x plus 12, factors into the quantity x plus 2 times the quantity x plus 6.", caption: none) #figure(figph[Step 2 is to simplify the rational expression, the quantity x plus 2 times the quantity x plus 3 all divided by the quantity x plus 2 times the quantity x plus 6, by dividing out the common factor, x plus 6. The result of removing the common factor is the quantity x plus 3 divided by the quantity x plus 6, where x is not equal to 2 and x is not equal to -6.], alt: "Step 2 is to simplify the rational expression, the quantity x plus 2 times the quantity x plus 3 all divided by the quantity x plus 2 times the quantity x plus 6, by dividing out the common factor, x plus 6. The result of removing the common factor is the quantity x plus 3 divided by the quantity x plus 6, where x is not equal to 2 and x is not equal to -6.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x squared minus x minus 2 over x squared minus 3 x plus 2 .")[$frac(x^(2) − x − 2, x^(2) − 3 x + 2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x plus 1 over x minus 1 ,")[$frac(x + 1, x − 1) ,$]#math.equation(block: true, alt: "x not equal to 2 ,")[$x ≠ 2 ,$]#math.equation(block: true, alt: "x not equal to 1")[$x ≠ 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x squared minus 3 x minus 10 over x squared plus x minus 2 .")[$frac(x^(2) − 3 x − 10, x^(2) + x − 2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x minus 5 over x minus 1 ,")[$frac(x − 5, x − 1) ,$]#math.equation(block: true, alt: "x not equal to − 2 ,")[$x ≠ "−" 2 ,$]#math.equation(block: true, alt: "x not equal to 1")[$x ≠ 1$] ] ] We now summarize the steps you should follow to simplify rational expressions. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Simplify a rational expression.] + Factor the numerator and denominator completely. + Simplify by dividing out common factors. ] Usually, we leave the simplified rational expression in factored form. This way, it is easy to check that we have removed #emph[all] the common factors. We’ll use the methods we have learned to factor the polynomials in the numerators and denominators in the following examples. Every time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples. #examplebox("Example 3")[][ Simplify: #math.equation(block: false, alt: "the fraction 3 a squared minus 12 a b plus 12 b squared over 6 a squared minus 24 b squared")[$frac(3 a^(2) − 12 a b + 12 b^(2), 6 a^(2) − 24 b^(2))$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 3 a squared minus 12 a b plus 12 b squared over 6 a squared minus 24 b squared")[$frac(3 a^(2) − 12 a b + 12 b^(2), 6 a^(2) − 24 b^(2))$]]), [Factor the numerator and denominator, #linebreak() first factoring out the GCF.], [#math.equation(block: false, alt: "the fraction 3 open parenthesis a squared minus 4 a b plus 4 b squared close parenthesis over 6 open parenthesis a squared minus 4 b squared close parenthesis")[$frac(3 ( a^(2) − 4 a b + 4 b^(2) ), 6 ( a^(2) − 4 b^(2) ))$]], [], [#math.equation(block: false, alt: "the fraction 3 open parenthesis a minus 2 b close parenthesis open parenthesis a minus 2 b close parenthesis over 6 open parenthesis a plus 2 b close parenthesis open parenthesis a minus 2 b close parenthesis")[$frac(3 ( a − 2 b ) ( a − 2 b ), 6 ( a + 2 b ) ( a − 2 b ))$]], [Remove the common factors of #math.equation(block: false, alt: "a minus 2 b and 3 .")[$a − 2 b #h(0.2em) "and" #h(0.2em) 3 .$]], [#math.equation(block: false, alt: "the fraction 3 open parenthesis a minus 2 b close parenthesis open parenthesis a minus 2 b close parenthesis over 3 times 2 open parenthesis a plus 2 b close parenthesis open parenthesis a minus 2 b close parenthesis")[$frac(cancel(3) ( a − 2 b ) cancel(( a − 2 b )), cancel(3) · 2 ( a + 2 b ) cancel(( a − 2 b )))$]], [], [#math.equation(block: false, alt: "the fraction a minus 2 b over 2 open parenthesis a plus 2 b close parenthesis")[$frac(a − 2 b, 2 ( a + 2 b ))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 2 x squared minus 12 x y plus 18 y squared over 3 x squared minus 27 y squared")[$frac(2 x^(2) − 12 x y + 18 y^(2), 3 x^(2) − 27 y^(2))$]. #solutionbox[ #math.equation(block: true, alt: "the fraction 2 open parenthesis x minus 3 y close parenthesis over 3 open parenthesis x plus 3 y close parenthesis")[$frac(2 ( x − 3 y ), 3 ( x + 3 y ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 5 x squared minus 30 x y plus 25 y squared over 2 x squared minus 50 y squared")[$frac(5 x^(2) − 30 x y + 25 y^(2), 2 x^(2) − 50 y^(2))$]. #solutionbox[ #math.equation(block: true, alt: "the fraction 5 open parenthesis x minus y close parenthesis over 2 open parenthesis x plus 5 y close parenthesis")[$frac(5 ( x − y ), 2 ( x + 5 y ))$] ] ] Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of #emph[a] is #math.equation(block: false, alt: "− a")[$"−" a$] and #math.equation(block: false, alt: "− a equals −1 times a .")[$"−" a = −1 · a .$] The numerical fraction, say #math.equation(block: false, alt: "the fraction 7 over −7")[$frac(7, −7)$] simplifies to #math.equation(block: false, alt: "−1")[$−1$]. We also recognize that the numerator and denominator are opposites. The fraction #math.equation(block: false, alt: "the fraction a over − a")[$frac(a, "−" a)$], whose numerator and denominator are opposites also simplifies to #math.equation(block: false, alt: "−1")[$−1$]. #math.equation(block: true, alt: "Let’s look at the expression b minus a ., b minus a; Rewrite., − a plus b; Factor out –1 ., −1 open parenthesis a minus b close parenthesis")[$"Let’s look at the expression" #h(0.2em) b − a . & & & #h(2em) b − a \ "Rewrite." & & & #h(2em) "−" a + b \ "Factor out" #h(0.2em) –1 . & & & #h(2em) −1 ( a − b )$]This tells us that #math.equation(block: false, alt: "b minus a")[$b − a$] is the opposite of #math.equation(block: false, alt: "a minus b .")[$a − b .$] In general, we could write the opposite of #math.equation(block: false, alt: "a minus b")[$a − b$] as #math.equation(block: false, alt: "b minus a .")[$b − a .$] So the rational expression #math.equation(block: false, alt: "the fraction a minus b over b minus a")[$frac(a − b, b − a)$] simplifies to #math.equation(block: false, alt: "−1 .")[$−1 .$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Opposites in a Rational Expression] The opposite of #math.equation(block: false, alt: "a minus b")[$a − b$] is #math.equation(block: false, alt: "b minus a .")[$b − a .$] #math.equation(block: true, alt: "the fraction a minus b over b minus a equals −1 a not equal to b")[$frac(a − b, b − a) = −1 #h(0.5em) a ≠ b$]An expression and its opposite divide to #math.equation(block: false, alt: "−1 .")[$−1 .$] ] We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat #math.equation(block: false, alt: "a plus b")[$a + b$] and #math.equation(block: false, alt: "b plus a")[$b + a$] as opposites. Recall that in addition, order doesn’t matter so #math.equation(block: false, alt: "a plus b equals b plus a")[$a + b = b + a$]. So if #math.equation(block: false, alt: "a not equal to − b")[$a ≠ "−" b$], then #math.equation(block: false, alt: "the fraction a plus b over b plus a equals 1 .")[$frac(a + b, b + a) = 1 .$] #examplebox("Example 4")[][ Simplify: #math.equation(block: false, alt: "the fraction x squared minus 4 x minus 32 over 64 minus x squared .")[$frac(x^(2) − 4 x − 32, 64 − x^(2)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression displaying the fraction (x^2 - 4x - 32) over (64 - x^2).], alt: "A mathematical expression displaying the fraction (x^2 - 4x - 32) over (64 - x^2).", caption: none)]), [Factor the numerator and the denominator.], [#figure(figph[A mathematical fraction is shown. The numerator is (x-8)(x+4). The denominator is (8-x)(8+x).], alt: "A mathematical fraction is shown. The numerator is (x-8)(x+4). The denominator is (8-x)(8+x).", caption: none)], [Recognize the factors that are opposites.], [#figure(figph[A mathematical expression showing the simplification of a rational function. The term (x-8) in the numerator and (8-x) in the denominator are canceled out, introducing a factor of -1, which combines with an existing -1.], alt: "A mathematical expression showing the simplification of a rational function. The term (x-8) in the numerator and (8-x) in the denominator are canceled out, introducing a factor of -1, which combines with an existing -1.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing a negative fraction: minus, then a fraction bar, with (x + 4) in the numerator and (x + 8) in the denominator.], alt: "A mathematical expression showing a negative fraction: minus, then a fraction bar, with (x + 4) in the numerator and (x + 8) in the denominator.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x squared minus 4 x minus 5 over 25 minus x squared .")[$frac(x^(2) − 4 x − 5, 25 − x^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction x plus 1 over x plus 5")[$− frac(x + 1, x + 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x squared plus x minus 2 over 1 minus x squared .")[$frac(x^(2) + x − 2, 1 − x^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction x plus 2 over x plus 1")[$− frac(x + 2, x + 1)$] ] ] === Multiply Rational Expressions To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Multiplication of Rational Expressions] If #emph[p], #emph[q], #emph[r], and #emph[s] are polynomials where #math.equation(block: false, alt: "q not equal to 0 , s not equal to 0 ,")[$q ≠ 0 , s ≠ 0 ,$] then #math.equation(block: true, alt: "the fraction p over q times the fraction r over s equals the fraction p r over q s")[$frac(p, q) · frac(r, s) = frac(p r, q s)$]To multiply rational expressions, multiply the numerators and multiply the denominators. ] Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, #math.equation(block: false, alt: "x not equal to 0 ,")[$x ≠ 0 ,$]#math.equation(block: false, alt: "x not equal to 3 ,")[$x ≠ 3 ,$] and #math.equation(block: false, alt: "x not equal to 4 .")[$x ≠ 4 .$] #examplebox("Example 5")[How to Multiply Rational Expressions][ Simplify: #math.equation(block: false, alt: "the fraction 2 x over x squared minus 7 x plus 12 times the fraction x squared minus 9 over 6 x squared .")[$frac(2 x, x^(2) − 7 x + 12) · frac(x^(2) − 9, 6 x^(2)) .$] #solutionbox[ #figure(figph[Step 1 is to factor each numerator and the denominator completely in 2 x divided by the quantity x squared minus 7 x plus 12 times the rational expression the quantity x squared minus 9 divided by 6 x squared. The denominator, x squared minus 7 x plus 12, factors into the quantity x minus 3 times the quantity x minus 4. The numerator x squared minus 9 factors into the quantity x minus 3 times the quantity x plus 3.], alt: "Step 1 is to factor each numerator and the denominator completely in 2 x divided by the quantity x squared minus 7 x plus 12 times the rational expression the quantity x squared minus 9 divided by 6 x squared. The denominator, x squared minus 7 x plus 12, factors into the quantity x minus 3 times the quantity x minus 4. The numerator x squared minus 9 factors into the quantity x minus 3 times the quantity x plus 3.", caption: none) #figure(figph[Step 2 is to multiply the numerators 2 x and the quantity x minus 3 times the quantity x plus 3, and the denominators the quantity x minus 3 times the quantity x minus 4 and 6 x squared. It is helpful to write the monomials in the numerator and in the denominator. first.], alt: "Step 2 is to multiply the numerators 2 x and the quantity x minus 3 times the quantity x plus 3, and the denominators the quantity x minus 3 times the quantity x minus 4 and 6 x squared. It is helpful to write the monomials in the numerator and in the denominator. first.", caption: none) #figure(figph[Step 3 is to simplify 2 x times the quantity x minus 3 times the quantity x plus 3 all divided by 2 times 3 times x times x times the quantity x minus 3 times the quantity x plus 4 by dividing out the common factor, x minus 3. Leaving the denominator in factored form, the result is the quantity x plus 3 divided by 3 x times the quantity x minus 4.], alt: "Step 3 is to simplify 2 x times the quantity x minus 3 times the quantity x plus 3 all divided by 2 times 3 times x times x times the quantity x minus 3 times the quantity x plus 4 by dividing out the common factor, x minus 3. Leaving the denominator in factored form, the result is the quantity x plus 3 divided by 3 x times the quantity x minus 4.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 5 x over x squared plus 5 x plus 6 times the fraction x squared minus 4 over 10 x .")[$frac(5 x, x^(2) + 5 x + 6) · frac(x^(2) − 4, 10 x) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x minus 2 over 2 open parenthesis x plus 3 close parenthesis")[$frac(x − 2, 2 ( x + 3 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 9 x squared over x squared plus 11 x plus 30 times the fraction x squared minus 36 over 3 x squared .")[$frac(9 x^(2), x^(2) + 11 x + 30) · frac(x^(2) − 36, 3 x^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 open parenthesis x minus 6 close parenthesis over x plus 5")[$frac(3 ( x − 6 ), x + 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Multiply rational expressions.] + Factor each numerator and denominator completely. + Multiply the numerators and denominators. + Simplify by dividing out common factors. ] #examplebox("Example 6")[][ Multiply: #math.equation(block: false, alt: "the fraction 3 a squared minus 8 a minus 3 over a squared minus 25 times the fraction a squared plus 10 a plus 25 over 3 a squared minus 14 a minus 5 .")[$frac(3 a^(2) − 8 a − 3, a^(2) − 25) · frac(a^(2) + 10 a + 25, 3 a^(2) − 14 a − 5) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 3 a squared minus 8 a minus 3 over a squared minus 25 times the fraction a squared plus 10 a plus 25 over 3 a squared minus 14 a minus 5")[$#h(2em) frac(3 a^(2) − 8 a − 3, a^(2) − 25) · frac(a^(2) + 10 a + 25, 3 a^(2) − 14 a − 5)$]]), [Factor the numerators and denominators #linebreak() and then multiply.], [#math.equation(block: false, alt: "the fraction open parenthesis 3 a plus 1 close parenthesis open parenthesis a minus 3 close parenthesis open parenthesis a plus 5 close parenthesis open parenthesis a plus 5 close parenthesis over open parenthesis a minus 5 close parenthesis open parenthesis a plus 5 close parenthesis open parenthesis 3 a plus 1 close parenthesis open parenthesis a minus 5 close parenthesis")[$#h(2em) frac(( 3 a + 1 ) ( a − 3 ) ( a + 5 ) ( a + 5 ), ( a − 5 ) ( a + 5 ) ( 3 a + 1 ) ( a − 5 ))$]], [Simplify by dividing out #linebreak() common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis 3 a plus 1 close parenthesis open parenthesis a minus 3 close parenthesis open parenthesis a plus 5 close parenthesis open parenthesis a plus 5 close parenthesis over open parenthesis a minus 5 close parenthesis open parenthesis a plus 5 close parenthesis open parenthesis 3 a plus 1 close parenthesis open parenthesis a minus 5 close parenthesis")[$#h(2em) frac(cancel(( 3 a + 1 )) ( a − 3 ) cancel(( a + 5 )) ( a + 5 ), ( a − 5 ) cancel(( a + 5 )) cancel(( 3 a + 1 )) ( a − 5 ))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction open parenthesis a minus 3 close parenthesis open parenthesis a plus 5 close parenthesis over open parenthesis a minus 5 close parenthesis open parenthesis a minus 5 close parenthesis")[$#h(2em) frac(( a − 3 ) ( a + 5 ), ( a − 5 ) ( a − 5 ))$]], [Rewrite #math.equation(block: false, alt: "open parenthesis a minus 5 close parenthesis open parenthesis a minus 5 close parenthesis")[$( a − 5 ) ( a − 5 )$] using an exponent.], [#math.equation(block: false, alt: "the fraction open parenthesis a minus 3 close parenthesis open parenthesis a plus 5 close parenthesis over open parenthesis a minus 5 close parenthesis squared")[$#h(2em) frac(( a − 3 ) ( a + 5 ), attach(( a − 5 ), t: 2))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 2 x squared plus 5 x minus 12 over x squared minus 16 times the fraction x squared minus 8 x plus 16 over 2 x squared minus 13 x plus 15 .")[$frac(2 x^(2) + 5 x − 12, x^(2) − 16) · frac(x^(2) − 8 x + 16, 2 x^(2) − 13 x + 15) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x minus 4 over x minus 5")[$frac(x − 4, x − 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 4 b squared plus 7 b minus 2 over 1 minus b squared times the fraction b squared minus 2 b plus 1 over 4 b squared plus 15 b minus 4 .")[$frac(4 b^(2) + 7 b − 2, 1 − b^(2)) · frac(b^(2) − 2 b + 1, 4 b^(2) + 15 b − 4) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction open parenthesis b plus 2 close parenthesis open parenthesis b minus 1 close parenthesis over open parenthesis 1 plus b close parenthesis open parenthesis b plus 4 close parenthesis")[$− frac(( b + 2 ) ( b − 1 ), ( 1 + b ) ( b + 4 ))$] ] ] === Divide Rational Expressions Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Division of Rational Expressions] If #emph[p], #emph[q], #emph[r,] and #emph[s] are polynomials where #math.equation(block: false, alt: "q not equal to 0 , r not equal to 0 , s not equal to 0 ,")[$q ≠ 0 , r ≠ 0 , s ≠ 0 ,$] then #math.equation(block: true, alt: "the fraction p over q divided by the fraction r over s equals the fraction p over q times the fraction s over r")[$frac(p, q) ÷ frac(r, s) = frac(p, q) · frac(s, r)$]To divide rational expressions, multiply the first fraction by the reciprocal of the second. ] Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors. #examplebox("Example 7")[How to Divide Rational Expressions][ Divide: #math.equation(block: false, alt: "the fraction p cubed plus q cubed over 2 p squared plus 2 p q plus 2 q squared divided by the fraction p squared minus q squared over 6 .")[$frac(p^(3) + q^(3), 2 p^(2) + 2 p q + 2 q^(2)) ÷ frac(p^(2) − q^(2), 6) .$] #solutionbox[ #figure(figph[Step 1 is to rewrite the division of the rational expression, the quantity p cubed plus q cubes divided by the quantity 2 p squared plus 2 p q plus 2 q squared divided by the rational expression, the quantity p squared minus q squared all divided by 6. Do this by flipping the rational expression, the quantity p squared minus q squared all divided by 6, and changing division to multiplication. The result is the quantity p cubed plus q cubes divided by the quantity 2 p squared plus 2 p q plus 2 q squared times the quantity 6 divided by the quantity p squared minus q squared.], alt: "Step 1 is to rewrite the division of the rational expression, the quantity p cubed plus q cubes divided by the quantity 2 p squared plus 2 p q plus 2 q squared divided by the rational expression, the quantity p squared minus q squared all divided by 6. Do this by flipping the rational expression, the quantity p squared minus q squared all divided by 6, and changing division to multiplication. The result is the quantity p cubed plus q cubes divided by the quantity 2 p squared plus 2 p q plus 2 q squared times the quantity 6 divided by the quantity p squared minus q squared.", caption: none) #figure(figph[Step 2 is to factor the numerators, the quantity p cubed plus q cubed and 6, and the denominators, the quantity 2 p squared plus 2 p q plus 2 squared and the quantity p squared minus q squared, completely. The result is the quantity p plus q times the quantity p squared minus p q plus q squared all times the quantity 2 times 3 divided by the quantity p minus q times the quantity p plus q.], alt: "Step 2 is to factor the numerators, the quantity p cubed plus q cubed and 6, and the denominators, the quantity 2 p squared plus 2 p q plus 2 squared and the quantity p squared minus q squared, completely. The result is the quantity p plus q times the quantity p squared minus p q plus q squared all times the quantity 2 times 3 divided by the quantity p minus q times the quantity p plus q.", caption: none) #figure(figph[Step 3 is to multiply the numerators and denominators. The result is the quantity p plus q times the quantity p squared minus p q plus q squared times 2 times 3 all divided by the 2 times the quantity p squared plus p q plus q squared times the quantity p minus q times the quantity p plus q.], alt: "Step 3 is to multiply the numerators and denominators. The result is the quantity p plus q times the quantity p squared minus p q plus q squared times 2 times 3 all divided by the 2 times the quantity p squared plus p q plus q squared times the quantity p minus q times the quantity p plus q.", caption: none) #figure(figph[Step 4 is to simplify the expression by dividing out the common factors, the quantity p plus q and 2. The result is 3 times the quantity p squared minus p q plus q squared all divided by the quantity p minus q times the quantity p squared plus p q plus q squared.], alt: "Step 4 is to simplify the expression by dividing out the common factors, the quantity p plus q and 2. The result is 3 times the quantity p squared minus p q plus q squared all divided by the quantity p minus q times the quantity p squared plus p q plus q squared.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x cubed plus 8 over 3 x squared minus 6 x plus 12 divided by the fraction x squared minus 4 over 6 .")[$frac(x^(3) + 8, 3 x^(2) − 6 x + 12) ÷ frac(x^(2) − 4, 6) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over x minus 2")[$frac(2, x − 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 2 z squared over z squared minus 1 divided by the fraction z cubed minus z squared plus z over z cubed plus 1 .")[$frac(2 z^(2), z^(2) − 1) ÷ frac(z^(3) − z^(2) + z, z^(3) + 1) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 z over z minus 1")[$frac(2 z, z − 1)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Divide rational expressions.] + Rewrite the division as the product of the first rational expression and the reciprocal of the second. + Factor the numerators and denominators completely. + Multiply the numerators and denominators together. + Simplify by dividing out common factors. ] Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions. #examplebox("Example 8")[][ Divide: #math.equation(block: false, alt: "the fraction the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 over the fraction 2 x squared minus 7 x plus 3 over x squared minus 5 x plus 6 .")[$frac(frac(6 x^(2) − 7 x + 2, 4 x − 8), frac(2 x^(2) − 7 x + 3, x^(2) − 5 x + 6)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 over the fraction 2 x squared minus 7 x plus 3 over x squared minus 5 x plus 6")[$frac(frac(6 x^(2) − 7 x + 2, 4 x − 8), frac(2 x^(2) − 7 x + 3, x^(2) − 5 x + 6))$]]), [Rewrite with a division sign.], [#math.equation(block: false, alt: "the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 divided by the fraction 2 x squared minus 7 x plus 3 over x squared minus 5 x plus 6")[$frac(6 x^(2) − 7 x + 2, 4 x − 8) ÷ frac(2 x^(2) − 7 x + 3, x^(2) − 5 x + 6)$]], [Rewrite as product of first times reciprocal #linebreak() of second.], [#math.equation(block: false, alt: "the fraction 6 x squared minus 7 x plus 2 over 4 x minus 8 times the fraction x squared minus 5 x plus 6 over 2 x squared minus 7 x plus 3")[$frac(6 x^(2) − 7 x + 2, 4 x − 8) · frac(x^(2) − 5 x + 6, 2 x^(2) − 7 x + 3)$]], [Factor the numerators and the #linebreak() denominators, and then multiply.], [#math.equation(block: false, alt: "the fraction open parenthesis 2 x minus 1 close parenthesis open parenthesis 3 x minus 2 close parenthesis open parenthesis x minus 2 close parenthesis open parenthesis x minus 3 close parenthesis over 4 open parenthesis x minus 2 close parenthesis open parenthesis 2 x minus 1 close parenthesis open parenthesis x minus 3 close parenthesis")[$frac(( 2 x − 1 ) ( 3 x − 2 ) ( x − 2 ) ( x − 3 ), 4 ( x − 2 ) ( 2 x − 1 ) ( x − 3 ))$]], [Simplify by dividing out common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis 2 x minus 1 close parenthesis open parenthesis 3 x minus 2 close parenthesis open parenthesis x minus 2 close parenthesis open parenthesis x minus 3 close parenthesis over 4 open parenthesis x minus 2 close parenthesis open parenthesis 2 x minus 1 close parenthesis open parenthesis x minus 3 close parenthesis")[$frac(cancel(( 2 x − 1 )) ( 3 x − 2 ) cancel(( x − 2 )) cancel(( x − 3 )), 4 cancel(( x − 2 )) cancel(( 2 x − 1 )) cancel(( x − 3 )))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 3 x minus 2 over 4")[$frac(3 x − 2, 4)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction the fraction 3 x squared plus 7 x plus 2 over 4 x plus 24 over the fraction 3 x squared minus 14 x minus 5 over x squared plus x minus 30 .")[$frac(frac(3 x^(2) + 7 x + 2, 4 x + 24), frac(3 x^(2) − 14 x − 5, x^(2) + x − 30)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x plus 2 over 4")[$frac(x + 2, 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction the fraction y squared minus 36 over 2 y squared plus 11 y minus 6 over the fraction 2 y squared minus 2 y minus 60 over 8 y minus 4 .")[$frac(frac(y^(2) − 36, 2 y^(2) + 11 y − 6), frac(2 y^(2) − 2 y − 60, 8 y − 4)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over y plus 5")[$frac(2, y + 5)$] ] ] If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply. #examplebox("Example 9")[][ Perform the indicated operations: #math.equation(block: false, alt: "the fraction 3 x minus 6 over 4 x minus 4 times the fraction x squared plus 2 x minus 3 over x squared minus 3 x minus 10 divided by the fraction 2 x plus 12 over 8 x plus 16 .")[$frac(3 x − 6, 4 x − 4) · frac(x^(2) + 2 x − 3, x^(2) − 3 x − 10) ÷ frac(2 x + 12, 8 x + 16) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical problem showing the multiplication and division of three rational algebraic expressions: (3x-6)/(4x-4) \* (x^2+2x-3)/(x^2-3x-10) / (2x+12)/(8x+16).], alt: "A mathematical problem showing the multiplication and division of three rational algebraic expressions: (3x-6)/(4x-4) * (x^2+2x-3)/(x^2-3x-10) / (2x+12)/(8x+16).", caption: none)]), [Rewrite the division as multiplication #linebreak() by the reciprocal.], [#figure(figph[A multiplication problem involving three rational algebraic expressions: (3x-6)/(4x-4), (x^2+2x-3)/(x^2-3x-10), and (8x+16)/(2x+12). The last expression is in red.], alt: "A multiplication problem involving three rational algebraic expressions: (3x-6)/(4x-4), (x^2+2x-3)/(x^2-3x-10), and (8x+16)/(2x+12). The last expression is in red.", caption: none)], [Factor the numerators and the denominators.], [#figure(figph[An algebraic expression showing the product of three rational functions: (3(x-2) / 4(x-1)) multiplied by ((x+3)(x-1) / (x+2)(x-5)) multiplied by (8(x+2) / 2(x+6)).], alt: "An algebraic expression showing the product of three rational functions: (3(x-2) / 4(x-1)) multiplied by ((x+3)(x-1) / (x+2)(x-5)) multiplied by (8(x+2) / 2(x+6)).", caption: none)], [Multiply the fractions. Bringing the constants to #linebreak() the front will help when removing common factors.], [], [Simplify by dividing out common factors.], [#figure(figph[Simplification of a rational algebraic expression, showing common factors (x-1) and (x+2) being cancelled from both the numerator and denominator, along with numerical terms 8, 4, and 2.], alt: "Simplification of a rational algebraic expression, showing common factors (x-1) and (x+2) being cancelled from both the numerator and denominator, along with numerical terms 8, 4, and 2.", caption: none)], [Simplify.], [#figure(figph[An algebraic fraction with 3(x-2)(x+3) in the numerator and (x-5)(x+6) in the denominator.], alt: "An algebraic fraction with 3(x-2)(x+3) in the numerator and (x-5)(x+6) in the denominator.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Perform the indicated operations: #math.equation(block: false, alt: "the fraction 4 m plus 4 over 3 m minus 15 times the fraction m squared minus 3 m minus 10 over m squared minus 4 m minus 32 divided by the fraction 12 m minus 36 over 6 m minus 48 .")[$frac(4 m + 4, 3 m − 15) · frac(m^(2) − 3 m − 10, m^(2) − 4 m − 32) ÷ frac(12 m − 36, 6 m − 48) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 open parenthesis m plus 1 close parenthesis open parenthesis m plus 2 close parenthesis over 3 open parenthesis m plus 4 close parenthesis open parenthesis m minus 3 close parenthesis")[$frac(2 ( m + 1 ) ( m + 2 ), 3 ( m + 4 ) ( m − 3 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Perform the indicated operations: #math.equation(block: false, alt: "the fraction 2 n squared plus 10 n over n minus 1 divided by the fraction n squared plus 10 n plus 24 over n squared plus 8 n minus 9 times the fraction n plus 4 over 8 n squared plus 12 n .")[$frac(2 n^(2) + 10 n, n − 1) ÷ frac(n^(2) + 10 n + 24, n^(2) + 8 n − 9) · frac(n + 4, 8 n^(2) + 12 n) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction open parenthesis n plus 5 close parenthesis open parenthesis n plus 9 close parenthesis over 2 open parenthesis n plus 6 close parenthesis open parenthesis 2 n plus 3 close parenthesis")[$frac(( n + 5 ) ( n + 9 ), 2 ( n + 6 ) ( 2 n + 3 ))$] ] ] === Multiply and Divide Rational Functions We started this section stating that a #strong[rational expression] is an expression of the form #math.equation(block: false, alt: "the fraction p over q ,")[$frac(p, q) ,$] where #emph[p] and #emph[q] are polynomials and #math.equation(block: false, alt: "q not equal to 0 .")[$q ≠ 0 .$] Similarly, we define a #strong[rational function] as a function of the form #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction p open parenthesis x close parenthesis over q open parenthesis x close parenthesis")[$R ( x ) = frac(p ( x ), q ( x ))$] where #math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$] and #math.equation(block: false, alt: "q open parenthesis x close parenthesis")[$q ( x )$] are polynomial functions and #math.equation(block: false, alt: "q open parenthesis x close parenthesis")[$q ( x )$] is not zero. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Rational Function] A rational function is a function of the form #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction p open parenthesis x close parenthesis over q open parenthesis x close parenthesis")[$R ( x ) = frac(p ( x ), q ( x ))$]where #math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$] and #math.equation(block: false, alt: "q open parenthesis x close parenthesis")[$q ( x )$] are polynomial functions and #math.equation(block: false, alt: "q open parenthesis x close parenthesis")[$q ( x )$] is not zero. ] The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make #math.equation(block: false, alt: "q open parenthesis x close parenthesis equals 0 .")[$q ( x ) = 0 .$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Determine the domain of a rational function.] + Set the denominator equal to zero. + Solve the equation. + The domain is all real numbers excluding the values found in Step 2. ] #examplebox("Example 10")[][ Find the domain of #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 2 x squared minus 14 x over 4 x squared minus 16 x minus 48 .")[$R ( x ) = frac(2 x^(2) − 14 x, 4 x^(2) − 16 x − 48) .$] #solutionbox[ The domain will be all real numbers except those values that make the denominator zero. We will set the denominator equal to zero , solve that equation, and then exclude those values from the domain. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Set the denominator to zero.], [#math.equation(block: false, alt: "4 x squared minus 16 x minus 48 equals 0")[$4 x^(2) − 16 x − 48 = 0 #h(0.7em)$]]), [Factor, first factor out the GCF.], [#math.equation(block: false, alt: "4 open parenthesis x squared minus 4 x minus 12 close parenthesis equals 0")[$4 ( x^(2) − 4 x − 12 ) = 0 #h(0.7em)$]], [], [#math.equation(block: false, alt: "4 open parenthesis x minus 6 close parenthesis open parenthesis x plus 2 close parenthesis equals 0")[$4 ( x − 6 ) ( x + 2 ) = 0 #h(0.7em)$]], [Use the Zero Product Property.], [#math.equation(block: false, alt: "4 not equal to 0 x minus 6 equals 0 x plus 2 equals 0")[$4 ≠ 0 #h(1em) x − 6 = 0 #h(1em) x + 2 = 0 #h(0.7em)$]], [Solve.], [#math.equation(block: false, alt: "x equals 6 x equals −2")[$x = 6 #h(1em) x = −2$]], [], [The domain of #math.equation(block: false, alt: "R open parenthesis x close parenthesis")[$R ( x )$] is all real numbers #linebreak() where #math.equation(block: false, alt: "x not equal to 6")[$x ≠ 6$] and #math.equation(block: false, alt: "x not equal to − 2 .")[$x ≠ "−" 2 .$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the domain of #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 2 x squared minus 10 x over 4 x squared minus 16 x minus 20 .")[$R ( x ) = frac(2 x^(2) − 10 x, 4 x^(2) − 16 x − 20) .$] #solutionbox[ The domain of #math.equation(block: false, alt: "R open parenthesis x close parenthesis")[$R ( x )$] is all real numbers where #math.equation(block: false, alt: "x not equal to 5")[$x ≠ 5$] and #math.equation(block: false, alt: "x not equal to − 1 .")[$x ≠ "−" 1 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the domain of #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 4 x squared minus 16 x over 8 x squared minus 16 x minus 64 .")[$R ( x ) = frac(4 x^(2) − 16 x, 8 x^(2) − 16 x − 64) .$] #solutionbox[ The domain of #math.equation(block: false, alt: "R open parenthesis x close parenthesis")[$R ( x )$] is all real numbers where #math.equation(block: false, alt: "x not equal to 4")[$x ≠ 4$] and #math.equation(block: false, alt: "x not equal to − 2 .")[$x ≠ "−" 2 .$] ] ] To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions. #examplebox("Example 11")[][ Find #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals f open parenthesis x close parenthesis times g open parenthesis x close parenthesis")[$R ( x ) = f ( x ) · g ( x )$] where #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 2 x minus 6 over x squared minus 8 x plus 15")[$f ( x ) = frac(2 x − 6, x^(2) − 8 x + 15)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction x squared minus 25 over 2 x plus 10 .")[$g ( x ) = frac(x^(2) − 25, 2 x + 10) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals f open parenthesis x close parenthesis times g open parenthesis x close parenthesis")[$#h(2em) R ( x ) = f ( x ) · g ( x )$]]), [], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 2 x minus 6 over x squared minus 8 x plus 15 times the fraction x squared minus 25 over 2 x plus 10")[$#h(2em) R ( x ) = frac(2 x − 6, x^(2) − 8 x + 15) · frac(x^(2) − 25, 2 x + 10)$]], [Factor each numerator and denominator.], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 2 open parenthesis x minus 3 close parenthesis over open parenthesis x minus 3 close parenthesis open parenthesis x minus 5 close parenthesis times the fraction open parenthesis x minus 5 close parenthesis open parenthesis x plus 5 close parenthesis over 2 open parenthesis x plus 5 close parenthesis")[$#h(2em) R ( x ) = frac(2 ( x − 3 ), ( x − 3 ) ( x − 5 )) · frac(( x − 5 ) ( x + 5 ), 2 ( x + 5 ))$]], [Multiply the numerators and denominators.], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 2 open parenthesis x minus 3 close parenthesis open parenthesis x minus 5 close parenthesis open parenthesis x plus 5 close parenthesis over 2 open parenthesis x minus 3 close parenthesis open parenthesis x minus 5 close parenthesis open parenthesis x plus 5 close parenthesis")[$#h(2em) R ( x ) = frac(2 ( x − 3 ) ( x − 5 ) ( x + 5 ), 2 ( x − 3 ) ( x − 5 ) ( x + 5 ))$]], [Remove common factors.], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 2 open parenthesis x minus 3 close parenthesis open parenthesis x minus 5 close parenthesis open parenthesis x plus 5 close parenthesis over 2 open parenthesis x minus 3 close parenthesis open parenthesis x minus 5 close parenthesis open parenthesis x plus 5 close parenthesis")[$#h(2em) R ( x ) = frac(cancel(2) cancel(( x − 3 )) cancel(( x − 5 )) cancel(( x + 5 )), cancel(2) cancel(( x − 3 )) cancel(( x − 5 )) cancel(( x + 5 )))$]], [Simplify.], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals 1")[$#h(2em) R ( x ) = 1$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals f open parenthesis x close parenthesis times g open parenthesis x close parenthesis")[$R ( x ) = f ( x ) · g ( x )$] where #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 3 x minus 21 over x squared minus 9 x plus 14")[$f ( x ) = frac(3 x − 21, x^(2) − 9 x + 14)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 2 x squared minus 8 over 3 x plus 6 .")[$g ( x ) = frac(2 x^(2) − 8, 3 x + 6) .$] #solutionbox[ #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals 2")[$R ( x ) = 2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals f open parenthesis x close parenthesis times g open parenthesis x close parenthesis")[$R ( x ) = f ( x ) · g ( x )$] where #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction x squared minus x over 3 x squared plus 27 x minus 30")[$f ( x ) = frac(x^(2) − x, 3 x^(2) + 27 x − 30)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction x squared minus 100 over x squared minus 10 x .")[$g ( x ) = frac(x^(2) − 100, x^(2) − 10 x) .$] #solutionbox[ #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction 1 over 3")[$R ( x ) = frac(1, 3)$] ] ] To divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions. #examplebox("Example 12")[][ Find #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction f open parenthesis x close parenthesis over g open parenthesis x close parenthesis")[$R ( x ) = frac(f ( x ), g ( x ))$] where #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 3 x squared over x squared minus 4 x")[$f ( x ) = frac(3 x^(2), x^(2) − 4 x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 9 x squared minus 45 x over x squared minus 7 x plus 10 .")[$g ( x ) = frac(9 x^(2) − 45 x, x^(2) − 7 x + 10) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction f open parenthesis x close parenthesis over g open parenthesis x close parenthesis")[$#h(2em) R ( x ) = frac(f ( x ), g ( x ))$]]), [Substitute in the functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis , g open parenthesis x close parenthesis .")[$f ( x ) , #h(0.2em) g ( x ) .$]], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction the fraction 3 x squared over x squared minus 4 x over the fraction 9 x squared minus 45 x over x squared minus 7 x plus 10")[$#h(2em) R ( x ) = frac(frac(3 x^(2), x^(2) − 4 x), frac(9 x^(2) − 45 x, x^(2) − 7 x + 10))$]], [Rewrite the division as the product of #linebreak() #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and the reciprocal of #math.equation(block: false, alt: "g open parenthesis x close parenthesis .")[$g ( x ) .$]], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 3 x squared over x squared minus 4 x times the fraction x squared minus 7 x plus 10 over 9 x squared minus 45 x")[$#h(2em) R ( x ) = frac(3 x^(2), x^(2) − 4 x) · frac(x^(2) − 7 x + 10, 9 x^(2) − 45 x)$]], [Factor the numerators and denominators #linebreak() and then multiply.], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 3 times x times x times open parenthesis x minus 5 close parenthesis open parenthesis x minus 2 close parenthesis over x open parenthesis x minus 4 close parenthesis times 3 times 3 times x times open parenthesis x minus 5 close parenthesis")[$#h(2em) R ( x ) = frac(3 · x · x · ( x − 5 ) ( x − 2 ), x ( x − 4 ) · 3 · 3 · x · ( x − 5 ))$]], [Simplify by dividing out common factors.], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction 3 times x times x open parenthesis x minus 5 close parenthesis open parenthesis x minus 2 close parenthesis over x open parenthesis x minus 4 close parenthesis times 3 times 3 times x open parenthesis x minus 5 close parenthesis")[$#h(2em) R ( x ) = frac(cancel(3) · cancel(x) · cancel(x) cancel(( x − 5 )) ( x − 2 ), cancel(x) ( x − 4 ) · cancel(3) · 3 · cancel(x) cancel(( x − 5 )))$]], [], [#math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction x minus 2 over 3 open parenthesis x minus 4 close parenthesis")[$#h(2em) R ( x ) = frac(x − 2, 3 ( x − 4 ))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction f open parenthesis x close parenthesis over g open parenthesis x close parenthesis")[$R ( x ) = frac(f ( x ), g ( x ))$] where #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 2 x squared over x squared minus 8 x")[$f ( x ) = frac(2 x^(2), x^(2) − 8 x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 8 x squared plus 24 x over x squared plus x minus 6 .")[$g ( x ) = frac(8 x^(2) + 24 x, x^(2) + x − 6) .$] #solutionbox[ #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction x minus 2 over 4 open parenthesis x minus 8 close parenthesis")[$R ( x ) = frac(x − 2, 4 ( x − 8 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction f open parenthesis x close parenthesis over g open parenthesis x close parenthesis")[$R ( x ) = frac(f ( x ), g ( x ))$] where #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 15 x squared over 3 x squared plus 33 x")[$f ( x ) = frac(15 x^(2), 3 x^(2) + 33 x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 5 x minus 5 over x squared plus 9 x minus 22 .")[$g ( x ) = frac(5 x − 5, x^(2) + 9 x − 22) .$] #solutionbox[ #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction x open parenthesis x minus 2 close parenthesis over x minus 1")[$R ( x ) = frac(x ( x − 2 ), x − 1)$] ] ] === Key Concepts - #strong[Determine the values for which a rational expression is undefined.] + Set the denominator equal to zero. + Solve the equation. - #strong[Equivalent Fractions Property] #linebreak() If #emph[a], #emph[b], and #emph[c] are numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 ,")[$b ≠ 0 , c ≠ 0 ,$] then #math.equation(block: false, alt: "the fraction a over b equals the fraction a times c over b times c")[$frac(a, b) = frac(a · c, b · c)$] and #math.equation(block: false, alt: "the fraction a times c over b times c equals the fraction a over b .")[$frac(a · c, b · c) = frac(a, b) .$] - #strong[How to simplify a rational expression.] + Factor the numerator and denominator completely. + Simplify by dividing out common factors. - #strong[Opposites in a Rational Expression] #linebreak()      The opposite of #math.equation(block: false, alt: "a minus b")[$a − b$] is #math.equation(block: false, alt: "b minus a .")[$b − a .$] #linebreak()     #math.equation(block: false, alt: "the fraction a minus b over b minus a equals −1 a not equal to b")[$frac(a − b, b − a) = −1 #h(8em) a ≠ b$] #linebreak()     An expression and its opposite divide to #math.equation(block: false, alt: "−1 .")[$−1 .$] - #strong[Multiplication of Rational Expressions] #linebreak() If #emph[p], #emph[q], #emph[r], and #emph[s] are polynomials where #math.equation(block: false, alt: "q not equal to 0 , s not equal to 0 ,")[$q ≠ 0 , s ≠ 0 ,$] then #linebreak() #math.equation(block: false, alt: "the fraction p over q times the fraction r over s equals the fraction p r over q s")[$#h(8em) frac(p, q) · frac(r, s) = frac(p r, q s)$] - #strong[How to multiply rational expressions.] + Factor each numerator and denominator completely. + Multiply the numerators and denominators. + Simplify by dividing out common factors. - #strong[Division of Rational Expressions] #linebreak() If #emph[p], #emph[q], #emph[r], and #emph[s] are polynomials where #math.equation(block: false, alt: "q not equal to 0 , r not equal to 0 , s not equal to 0 ,")[$q ≠ 0 , r ≠ 0 , s ≠ 0 ,$] then #linebreak() #math.equation(block: false, alt: "the fraction p over q divided by the fraction r over s equals the fraction p over q times the fraction s over r")[$#h(8em) frac(p, q) ÷ frac(r, s) = frac(p, q) · frac(s, r)$] - #strong[How to divide rational expressions.] + Rewrite the division as the product of the first rational expression and the reciprocal of the second. + Factor the numerators and denominators completely. + Multiply the numerators and denominators together. + Simplify by dividing out common factors. - #strong[How to determine the domain of a rational function.] + Set the denominator equal to zero. + Solve the equation. + The domain is all real numbers excluding the values found in Step 2. ==== Practice Makes Perfect #strong[Determine the Values for Which a Rational Expression is Undefined] In the following exercises, determine the values for which the rational expression is undefined. ⓐ #math.equation(block: false, alt: "the fraction 2 x squared over z")[$frac(2 x^(2), z)$], ⓑ #math.equation(block: false, alt: "the fraction 4 p minus 1 over 6 p minus 5")[$frac(4 p − 1, 6 p − 5)$], ⓒ #math.equation(block: false, alt: "the fraction n minus 3 over n squared plus 2 n minus 8")[$frac(n − 3, n^(2) + 2 n − 8)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "z equals 0")[$z = 0$] ⓑ #math.equation(block: false, alt: "p equals the fraction 5 over 6")[$p = frac(5, 6)$] #linebreak() ⓒ #math.equation(block: false, alt: "n equals −4 , n equals 2")[$n = −4 , n = 2$] ] ⓐ #math.equation(block: false, alt: "the fraction 10 m over 11 n")[$frac(10 m, 11 n)$], ⓑ #math.equation(block: false, alt: "the fraction 6 y plus 13 over 4 y minus 9")[$frac(6 y + 13, 4 y − 9)$], ⓒ #math.equation(block: false, alt: "the fraction b minus 8 over b squared minus 36")[$frac(b − 8, b^(2) − 36)$] ⓐ #math.equation(block: false, alt: "the fraction 4 x squared y over 3 y")[$frac(4 x^(2) y, 3 y)$], ⓑ #math.equation(block: false, alt: "the fraction 3 x minus 2 over 2 x plus 1")[$frac(3 x − 2, 2 x + 1)$], ⓒ #math.equation(block: false, alt: "the fraction u minus 1 over u squared minus 3 u minus 28")[$frac(u − 1, u^(2) − 3 u − 28)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals 0")[$y = 0$], ⓑ #math.equation(block: false, alt: "x equals minus the fraction 1 over 2")[$x = − frac(1, 2)$], ⓒ #math.equation(block: false, alt: "u equals −4 , u equals 7")[$u = −4 , u = 7$] ] ⓐ #math.equation(block: false, alt: "the fraction 5 p q squared over 9 q")[$frac(5 p q^(2), 9 q)$], ⓑ #math.equation(block: false, alt: "the fraction 7 a minus 4 over 3 a plus 5")[$frac(7 a − 4, 3 a + 5)$], ⓒ #math.equation(block: false, alt: "the fraction 1 over x squared minus 4")[$frac(1, x^(2) − 4)$] #strong[Simplify Rational Expressions] In the following exercises, simplify each rational expression. #math.equation(block: true, alt: "minus the fraction 44 over 55")[$− frac(44, 55)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 over 5")[$− frac(4, 5)$] ] #math.equation(block: true, alt: "the fraction 56 over 63")[$frac(56, 63)$] #math.equation(block: true, alt: "the fraction 8 m cubed n over 12 m n squared")[$frac(8 m^(3) n, 12 m n^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 m squared over 3 n")[$frac(2 m^(2), 3 n)$] ] #math.equation(block: true, alt: "the fraction 36 v cubed w squared over 27 v w cubed")[$frac(36 v^(3) w^(2), 27 v w^(3))$] #math.equation(block: true, alt: "the fraction 8 n minus 96 over 3 n minus 36")[$frac(8 n − 96, 3 n − 36)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 8 over 3 open parenthesis n not equal to 2 close parenthesis")[$frac(8, 3) ( n ≠ 2 )$] ] #math.equation(block: true, alt: "the fraction 12 p minus 240 over 5 p minus 100")[$frac(12 p − 240, 5 p − 100)$] #math.equation(block: true, alt: "the fraction x squared plus 4 x minus 5 over x squared minus 2 x plus 1")[$frac(x^(2) + 4 x − 5, x^(2) − 2 x + 1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction x plus 5 over x minus 1")[$frac(x + 5, x − 1)$] ] #math.equation(block: true, alt: "the fraction y squared plus 3 y minus 4 over y squared minus 6 y plus 5")[$frac(y^(2) + 3 y − 4, y^(2) − 6 y + 5)$] #math.equation(block: true, alt: "the fraction a squared minus 4 over a squared plus 6 a minus 16")[$frac(a^(2) − 4, a^(2) + 6 a − 16)$] #solutionbox[ #math.equation(block: true, alt: "the fraction a plus 2 over a plus 8")[$frac(a + 2, a + 8)$] ] #math.equation(block: true, alt: "the fraction y squared minus 2 y minus 3 over y squared minus 9")[$frac(y^(2) − 2 y − 3, y^(2) − 9)$] #math.equation(block: true, alt: "the fraction p cubed plus 3 p squared plus 4 p plus 12 over p squared plus p minus 6")[$frac(p^(3) + 3 p^(2) + 4 p + 12, p^(2) + p − 6)$] #solutionbox[ #math.equation(block: true, alt: "the fraction p squared plus 4 over p minus 2")[$frac(p^(2) + 4, p − 2)$] ] #math.equation(block: true, alt: "the fraction x cubed minus 2 x squared minus 25 x plus 50 over x squared minus 25")[$frac(x^(3) − 2 x^(2) − 25 x + 50, x^(2) − 25)$] #math.equation(block: true, alt: "the fraction 8 b squared minus 32 b over 2 b squared minus 6 b minus 80")[$frac(8 b^(2) − 32 b, 2 b^(2) − 6 b − 80)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 b open parenthesis b minus 4 close parenthesis over open parenthesis b plus 5 close parenthesis open parenthesis b minus 8 close parenthesis")[$frac(4 b ( b − 4 ), ( b + 5 ) ( b − 8 ))$] ] #math.equation(block: true, alt: "the fraction −5 c squared minus 10 c over −10 c squared plus 30 c plus 100")[$frac(−5 c^(2) − 10 c, −10 c^(2) + 30 c + 100)$] #math.equation(block: true, alt: "the fraction 3 m squared plus 30 m n plus 75 n squared over 4 m squared minus 100 n squared")[$frac(3 m^(2) + 30 m n + 75 n^(2), 4 m^(2) − 100 n^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 open parenthesis m plus 5 n close parenthesis over 4 open parenthesis m minus 5 n close parenthesis")[$frac(3 ( m + 5 n ), 4 ( m − 5 n ))$] ] #math.equation(block: true, alt: "the fraction 5 r squared plus 30 r s minus 35 s squared over r squared minus 49 s squared")[$frac(5 r^(2) + 30 r s − 35 s^(2), r^(2) − 49 s^(2))$] #math.equation(block: true, alt: "the fraction a minus 5 over 5 minus a")[$frac(a − 5, 5 − a)$] #solutionbox[ #math.equation(block: true, alt: "−1")[$−1$] ] #math.equation(block: true, alt: "the fraction 5 minus d over d minus 5")[$frac(5 − d, d − 5)$] #math.equation(block: true, alt: "the fraction 20 minus 5 y over y squared minus 16")[$frac(20 − 5 y, y^(2) − 16)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over y plus 4")[$− frac(5, y + 4)$] ] #math.equation(block: true, alt: "the fraction 4 v minus 32 over 64 minus v squared")[$frac(4 v − 32, 64 − v^(2))$] #math.equation(block: true, alt: "the fraction w cubed plus 216 over w squared minus 36")[$frac(w^(3) + 216, w^(2) − 36)$] #solutionbox[ #math.equation(block: true, alt: "the fraction w squared minus 6 w plus 36 over w minus 6")[$frac(w^(2) − 6 w + 36, w − 6)$] ] #math.equation(block: true, alt: "the fraction v cubed plus 125 over v squared minus 25")[$frac(v^(3) + 125, v^(2) − 25)$] #math.equation(block: true, alt: "the fraction z squared minus 9 z plus 20 over 16 minus z squared")[$frac(z^(2) − 9 z + 20, 16 − z^(2))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction z minus 5 over 4 plus z")[$− frac(z − 5, 4 + z)$] ] #math.equation(block: true, alt: "the fraction a squared minus 5 a minus 36 over 81 minus a squared")[$frac(a^(2) − 5 a − 36, 81 − a^(2))$] #strong[Multiply Rational Expressions] In the following exercises, multiply the rational expressions. #math.equation(block: true, alt: "the fraction 12 over 16 times the fraction 4 over 10")[$frac(12, 16) · frac(4, 10)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 10")[$frac(3, 10)$] ] #math.equation(block: true, alt: "the fraction 32 over 5 times the fraction 16 over 24")[$frac(32, 5) · frac(16, 24)$] #math.equation(block: true, alt: "the fraction 5 x squared y to the power 4 over 12 x y cubed times the fraction 6 x squared over 20 y squared")[$frac(5 x^(2) y^(4), 12 x y^(3)) · frac(6 x^(2), 20 y^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction x cubed over 8 y")[$frac(x^(3), 8 y)$] ] #math.equation(block: true, alt: "the fraction 12 a cubed b over b squared times the fraction 2 a b squared over 9 b cubed")[$frac(12 a^(3) b, b^(2)) · frac(2 a b^(2), 9 b^(3))$] #math.equation(block: true, alt: "the fraction 5 p squared over p squared minus 5 p minus 36 times the fraction p squared minus 16 over 10 p")[$frac(5 p^(2), p^(2) − 5 p − 36) · frac(p^(2) − 16, 10 p)$] #solutionbox[ #math.equation(block: true, alt: "the fraction p open parenthesis p minus 4 close parenthesis over 2 open parenthesis p minus 9 close parenthesis")[$frac(p ( p − 4 ), 2 ( p − 9 ))$] ] #math.equation(block: true, alt: "the fraction 3 q squared over q squared plus q minus 6 times the fraction q squared minus 9 over 9 q")[$frac(3 q^(2), q^(2) + q − 6) · frac(q^(2) − 9, 9 q)$] #math.equation(block: true, alt: "the fraction 2 y squared minus 10 y over y squared plus 10 y plus 25 times the fraction y plus 5 over 6 y")[$frac(2 y^(2) − 10 y, y^(2) + 10 y + 25) · frac(y + 5, 6 y)$] #solutionbox[ #math.equation(block: true, alt: "the fraction y minus 5 over 3 open parenthesis y plus 5 close parenthesis")[$frac(y − 5, 3 ( y + 5 ))$] ] #math.equation(block: true, alt: "the fraction z squared plus 3 z over z squared minus 3 z minus 4 times the fraction z minus 4 over z squared")[$frac(z^(2) + 3 z, z^(2) − 3 z − 4) · frac(z − 4, z^(2))$] #math.equation(block: true, alt: "the fraction 28 minus 4 b over 3 b minus 3 times the fraction b squared plus 8 b minus 9 over b squared minus 49")[$frac(28 − 4 b, 3 b − 3) · frac(b^(2) + 8 b − 9, b^(2) − 49)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 open parenthesis b plus 9 close parenthesis over 3 open parenthesis b plus 7 close parenthesis")[$− frac(4 ( b + 9 ), 3 ( b + 7 ))$] ] #math.equation(block: true, alt: "the fraction 72 m minus 12 m squared over 8 m plus 32 times the fraction m squared plus 10 m plus 24 over m squared minus 36")[$frac(72 m − 12 m^(2), 8 m + 32) · frac(m^(2) + 10 m + 24, m^(2) − 36)$] #math.equation(block: true, alt: "the fraction 3 c squared minus 16 c plus 5 over c squared minus 25 times the fraction c squared plus 10 c plus 25 over 3 c squared minus 14 c minus 5")[$frac(3 c^(2) − 16 c + 5, c^(2) − 25) · frac(c^(2) + 10 c + 25, 3 c^(2) − 14 c − 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction open parenthesis 3 c minus 1 close parenthesis open parenthesis c plus 5 close parenthesis over open parenthesis 3 c plus 1 close parenthesis open parenthesis c minus 5 close parenthesis")[$frac(( 3 c − 1 ) ( c + 5 ), ( 3 c + 1 ) ( c − 5 ))$] ] #math.equation(block: true, alt: "the fraction 2 d squared plus d minus 3 over d squared minus 16 times the fraction d squared minus 8 d plus 16 over 2 d squared minus 9 d minus 18")[$frac(2 d^(2) + d − 3, d^(2) − 16) · frac(d^(2) − 8 d + 16, 2 d^(2) − 9 d − 18)$] #math.equation(block: true, alt: "the fraction 6 m squared minus 13 m plus 2 over 9 minus m squared times the fraction m squared minus 6 m plus 9 over 6 m squared plus 23 m minus 4")[$frac(6 m^(2) − 13 m + 2, 9 − m^(2)) · frac(m^(2) − 6 m + 9, 6 m^(2) + 23 m − 4)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction open parenthesis m minus 2 close parenthesis open parenthesis m minus 3 close parenthesis over open parenthesis 3 plus m close parenthesis open parenthesis m plus 4 close parenthesis")[$− frac(( m − 2 ) ( m − 3 ), ( 3 + m ) ( m + 4 ))$] ] #math.equation(block: true, alt: "the fraction 2 n squared minus 3 n minus 14 over 25 minus n squared times the fraction n squared minus 10 n plus 25 over 2 n squared minus 13 n plus 21")[$frac(2 n^(2) − 3 n − 14, 25 − n^(2)) · frac(n^(2) − 10 n + 25, 2 n^(2) − 13 n + 21)$] #strong[Divide Rational Expressions] In the following exercises, divide the rational expressions. #math.equation(block: true, alt: "the fraction v minus 5 over 11 minus v divided by the fraction v squared minus 25 over v minus 11")[$frac(v − 5, 11 − v) ÷ frac(v^(2) − 25, v − 11)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over v plus 5")[$− frac(1, v + 5)$] ] #math.equation(block: true, alt: "the fraction 10 plus w over w minus 8 divided by the fraction 100 minus w squared over 8 minus w")[$frac(10 + w, w − 8) ÷ frac(100 − w^(2), 8 − w)$] #math.equation(block: true, alt: "the fraction 3 s squared over s squared minus 16 divided by the fraction s cubed plus 4 s squared plus 16 s over s cubed minus 64")[$frac(3 s^(2), s^(2) − 16) ÷ frac(s^(3) + 4 s^(2) + 16 s, s^(3) − 64)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 s over s plus 4")[$frac(3 s, s + 4)$] ] #math.equation(block: true, alt: "the fraction r squared minus 9 over 15 divided by the fraction r cubed minus 27 over 5 r squared plus 15 r plus 45")[$frac(r^(2) − 9, 15) ÷ frac(r^(3) − 27, 5 r^(2) + 15 r + 45)$] #math.equation(block: true, alt: "the fraction p cubed plus q cubed over 3 p squared plus 3 p q plus 3 q squared divided by the fraction p squared minus q squared over 12")[$frac(p^(3) + q^(3), 3 p^(2) + 3 p q + 3 q^(2)) ÷ frac(p^(2) − q^(2), 12)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 open parenthesis p squared minus p q plus q squared close parenthesis over open parenthesis p minus q close parenthesis open parenthesis p squared plus p q plus q squared close parenthesis")[$frac(4 ( p^(2) − p q + q^(2) ), ( p − q ) ( p^(2) + p q + q^(2) ))$] ] #math.equation(block: true, alt: "the fraction v cubed minus 8 w cubed over 2 v squared plus 4 v w plus 8 w squared divided by the fraction v squared minus 4 w squared over 4")[$frac(v^(3) − 8 w^(3), 2 v^(2) + 4 v w + 8 w^(2)) ÷ frac(v^(2) − 4 w^(2), 4)$] #math.equation(block: true, alt: "the fraction x squared plus 3 x minus 10 over 4 x divided by open parenthesis 2 x squared plus 20 x plus 50 close parenthesis")[$frac(x^(2) + 3 x − 10, 4 x) ÷ ( 2 x^(2) + 20 x + 50 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction x minus 2 over 8 x open parenthesis x plus 5 close parenthesis")[$frac(x − 2, 8 x ( x + 5 ))$] ] #math.equation(block: true, alt: "the fraction 2 y squared minus 10 y z minus 48 z squared over 2 y minus 1 divided by open parenthesis 4 y squared minus 32 y z close parenthesis")[$frac(2 y^(2) − 10 y z − 48 z^(2), 2 y − 1) ÷ ( 4 y^(2) − 32 y z )$] #math.equation(block: true, alt: "the fraction the fraction 2 a squared minus a minus 21 over 5 a plus 20 over the fraction a squared plus 7 a plus 12 over a squared plus 8 a plus 16")[$frac(frac(2 a^(2) − a − 21, 5 a + 20), frac(a^(2) + 7 a + 12, a^(2) + 8 a + 16))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 a minus 7 over 5")[$frac(2 a − 7, 5)$] ] #math.equation(block: true, alt: "the fraction the fraction 3 b squared plus 2 b minus 8 over 12 b plus 18 over the fraction 3 b squared plus 2 b minus 8 over 2 b squared minus 7 b minus 15")[$frac(frac(3 b^(2) + 2 b − 8, 12 b + 18), frac(3 b^(2) + 2 b − 8, 2 b^(2) − 7 b − 15))$] #math.equation(block: true, alt: "the fraction the fraction 12 c squared minus 12 over 2 c squared minus 3 c plus 1 over the fraction 4 c plus 4 over 6 c squared minus 13 c plus 5")[$frac(frac(12 c^(2) − 12, 2 c^(2) − 3 c + 1), frac(4 c + 4, 6 c^(2) − 13 c + 5))$] #solutionbox[ #math.equation(block: true, alt: "3 open parenthesis 3 c minus 5 close parenthesis")[$3 ( 3 c − 5 )$] ] #math.equation(block: true, alt: "the fraction the fraction 4 d squared plus 7 d minus 2 over 35 d plus 10 over the fraction d squared minus 4 over 7 d squared minus 12 d minus 4")[$frac(frac(4 d^(2) + 7 d − 2, 35 d + 10), frac(d^(2) − 4, 7 d^(2) − 12 d − 4))$] For the following exercises, perform the indicated operations. #math.equation(block: true, alt: "the fraction 10 m squared plus 80 m over 3 m minus 9 times the fraction m squared plus 4 m minus 21 over m squared minus 9 m plus 20 divided by the fraction 5 m squared plus 10 m over 2 m minus 10")[$frac(10 m^(2) + 80 m, 3 m − 9) · frac(m^(2) + 4 m − 21, m^(2) − 9 m + 20) ÷ frac(5 m^(2) + 10 m, 2 m − 10)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 open parenthesis m plus 8 close parenthesis open parenthesis m plus 7 close parenthesis over 3 open parenthesis m minus 4 close parenthesis open parenthesis m plus 2 close parenthesis")[$frac(4 ( m + 8 ) ( m + 7 ), 3 ( m − 4 ) ( m + 2 ))$] ] #math.equation(block: true, alt: "the fraction 4 n squared plus 32 n over 3 n plus 2 times the fraction 3 n squared minus n minus 2 over n squared plus n minus 30 divided by the fraction 108 n squared minus 24 n over n plus 6")[$frac(4 n^(2) + 32 n, 3 n + 2) · frac(3 n^(2) − n − 2, n^(2) + n − 30) ÷ frac(108 n^(2) − 24 n, n + 6)$] #math.equation(block: true, alt: "the fraction 12 p squared plus 3 p over p plus 3 divided by the fraction p squared plus 2 p minus 63 over p squared minus p minus 12 times the fraction p minus 7 over 9 p cubed minus 9 p squared")[$frac(12 p^(2) + 3 p, p + 3) ÷ frac(p^(2) + 2 p − 63, p^(2) − p − 12) · frac(p − 7, 9 p^(3) − 9 p^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction open parenthesis 4 p plus 1 close parenthesis open parenthesis p minus 4 close parenthesis over 3 p open parenthesis p plus 9 close parenthesis open parenthesis p minus 1 close parenthesis")[$frac(( 4 p + 1 ) ( p − 4 ), 3 p ( p + 9 ) ( p − 1 ))$] ] #math.equation(block: true, alt: "the fraction 6 q plus 3 over 9 q squared minus 9 q divided by the fraction q squared plus 14 q plus 33 over q squared plus 4 q minus 5 times the fraction 4 q squared plus 12 q over 12 q plus 6")[$frac(6 q + 3, 9 q^(2) − 9 q) ÷ frac(q^(2) + 14 q + 33, q^(2) + 4 q − 5) · frac(4 q^(2) + 12 q, 12 q + 6)$] #strong[Multiply and Divide Rational Functions] In the following exercises, find the domain of each function. #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction x cubed minus 2 x squared minus 25 x plus 50 over x squared minus 25")[$R ( x ) = frac(x^(3) − 2 x^(2) − 25 x + 50, x^(2) − 25)$] #solutionbox[ #math.equation(block: false, alt: "x not equal to 5")[$x ≠ 5$] and #math.equation(block: false, alt: "x not equal to − 5")[$x ≠ "−" 5$] ] #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction x cubed plus 3 x squared minus 4 x minus 12 over x squared minus 4")[$R ( x ) = frac(x^(3) + 3 x^(2) − 4 x − 12, x^(2) − 4)$] #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction 3 x squared plus 15 x over 6 x squared plus 6 x minus 36")[$R ( x ) = frac(3 x^(2) + 15 x, 6 x^(2) + 6 x − 36)$] #solutionbox[ #math.equation(block: false, alt: "x not equal to 2")[$x ≠ 2$] and #math.equation(block: false, alt: "x not equal to − 3")[$x ≠ "−" 3$] ] #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction 8 x squared minus 32 x over 2 x squared minus 6 x minus 80")[$R ( x ) = frac(8 x^(2) − 32 x, 2 x^(2) − 6 x − 80)$] For the following exercises, find #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals f open parenthesis x close parenthesis times g open parenthesis x close parenthesis")[$R ( x ) = f ( x ) · g ( x )$] where #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] are given. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 6 x squared minus 12 x over x squared plus 7 x minus 18")[$f ( x ) = frac(6 x^(2) − 12 x, x^(2) + 7 x − 18)$] #linebreak() #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the fraction x squared minus 81 over 3 x squared minus 27 x")[$#h(1.4em) g ( x ) = frac(x^(2) − 81, 3 x^(2) − 27 x)$] #solutionbox[ #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals 2")[$R ( x ) = 2$] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction x squared minus 2 x over x squared plus 6 x minus 16")[$f ( x ) = frac(x^(2) − 2 x, x^(2) + 6 x − 16)$] #linebreak() #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the fraction x squared minus 64 over x squared minus 8 x")[$#h(1.4em) g ( x ) = frac(x^(2) − 64, x^(2) − 8 x)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 4 x over x squared minus 3 x minus 10")[$f ( x ) = frac(4 x, x^(2) − 3 x − 10)$] #linebreak() #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the fraction x squared minus 25 over 8 x squared")[$#h(1.4em) g ( x ) = frac(x^(2) − 25, 8 x^(2))$] #solutionbox[ #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction x plus 5 over 2 x open parenthesis x plus 2 close parenthesis")[$R ( x ) = frac(x + 5, 2 x ( x + 2 ))$] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 2 x squared plus 8 x over x squared minus 9 x plus 20")[$f ( x ) = frac(2 x^(2) + 8 x, x^(2) − 9 x + 20)$] #linebreak() #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the fraction x minus 5 over x squared")[$#h(1.4em) g ( x ) = frac(x − 5, x^(2))$] For the following exercises, find #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals the fraction f open parenthesis x close parenthesis over g open parenthesis x close parenthesis")[$R ( x ) = frac(f ( x ), g ( x ))$] where #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] are given. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 27 x squared over 3 x minus 21")[$f ( x ) = frac(27 x^(2), 3 x − 21)$] #linebreak() #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the fraction 3 x squared plus 18 x over x squared plus 13 x plus 42")[$#h(1.4em) g ( x ) = frac(3 x^(2) + 18 x, x^(2) + 13 x + 42)$] #solutionbox[ #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction 3 x open parenthesis x plus 7 close parenthesis over x minus 7")[$R ( x ) = frac(3 x ( x + 7 ), x − 7)$] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 24 x squared over 2 x minus 8")[$f ( x ) = frac(24 x^(2), 2 x − 8)$] #linebreak() #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the fraction 4 x cubed plus 28 x squared over x squared plus 11 x plus 28")[$#h(1.4em) g ( x ) = frac(4 x^(3) + 28 x^(2), x^(2) + 11 x + 28)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 16 x squared over 4 x plus 36")[$f ( x ) = frac(16 x^(2), 4 x + 36)$] #linebreak() #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the fraction 4 x squared minus 24 x over x squared plus 4 x minus 45")[$#h(1.4em) g ( x ) = frac(4 x^(2) − 24 x, x^(2) + 4 x − 45)$] #solutionbox[ #math.equation(block: true, alt: "R open parenthesis x close parenthesis equals the fraction x open parenthesis x minus 5 close parenthesis over x minus 6")[$R ( x ) = frac(x ( x − 5 ), x − 6)$] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 24 x squared over 2 x minus 4")[$f ( x ) = frac(24 x^(2), 2 x − 4)$] #linebreak() #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the fraction 12 x squared plus 36 x over x squared minus 11 x plus 18")[$#h(1.4em) g ( x ) = frac(12 x^(2) + 36 x, x^(2) − 11 x + 18)$] ==== Writing Exercises Explain how you find the values of #emph[x] for which the rational expression #math.equation(block: false, alt: "the fraction x squared minus x minus 20 over x squared minus 4")[$frac(x^(2) − x − 20, x^(2) − 4)$] is undefined. #solutionbox[ Answers will vary. ] Explain all the steps you take to simplify the rational expression #math.equation(block: false, alt: "the fraction p squared plus 4 p minus 21 over 9 minus p squared .")[$frac(p^(2) + 4 p − 21, 9 − p^(2)) .$] ⓐ Multiply #math.equation(block: false, alt: "the fraction 7 over 4 times the fraction 9 over 10")[$frac(7, 4) · frac(9, 10)$] and explain all your steps. ⓑ Multiply #math.equation(block: false, alt: "the fraction n over n minus 3 times the fraction 9 over n plus 3")[$frac(n, n − 3) · frac(9, n + 3)$] and explain all your steps. ⓒ Evaluate your answer to part ⓑ when #math.equation(block: false, alt: "n equals 7")[$n = 7$]. Did you get the same answer you got in part ⓐ ? Why or why not? #solutionbox[ Answers will vary. ] ⓐ Divide #math.equation(block: false, alt: "the fraction 24 over 5 divided by 6")[$frac(24, 5) ÷ 6$] and explain all your steps. ⓑ Divide #math.equation(block: false, alt: "the fraction x squared minus 1 over x divided by open parenthesis x plus 1 close parenthesis")[$frac(x^(2) − 1, x) ÷ ( x + 1 )$] and explain all your steps. ⓒ Evaluate your answer to part ⓑ when #math.equation(block: false, alt: "x equals 5 .")[$x = 5 .$] Did you get the same answer you got in part ⓐ ? Why or why not? ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This table has four columns and six rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was determine the values for which a rational expression is undefined. In row 3, the I can was simplify rationale expressions. In row 4, the I can was multiply rational expressions. In row 5, the I can was divide rational expressions. In row 6, the I can was multiply and divide rational functions. There is the nothing in the other columns.], alt: "This table has four columns and six rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was determine the values for which a rational expression is undefined. In row 3, the I can was simplify rationale expressions. In row 4, the I can was multiply rational expressions. In row 5, the I can was divide rational expressions. In row 6, the I can was multiply and divide rational functions. There is the nothing in the other columns.", caption: none) ⓑ If most of your checks were: #strong[…confidently.] Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific! #strong[…with some help.] This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? #strong[…no - I don’t get it!] This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.