#set document(title: "8.1 Simplify 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")) == 8.1#h(0.6em)Simplify Rational Expressions #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material. 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"))[ Factor: #math.equation(block: false, alt: "6 x squared minus 7 x plus 2 .")[$6 x^(2) − 7 x + 2 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "2 x minus 1 3 x minus 2")[$2 x − 1 3 x − 2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Factor: #math.equation(block: false, alt: "n cubed plus 8 .")[$n^(3) + 8 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "n plus 2 n squared minus 2 n plus 4")[$n + 2 n^(2) − 2 n + 4$] ] ] In Chapter 1, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Rational Expression] A #strong[rational expression] is an expression of the form #math.equation(block: false, alt: "the fraction p open parenthesis x close parenthesis over q open parenthesis x close parenthesis ,")[$frac(p ( x ), q ( x )) ,$] where #emph[p] and #emph[q] are polynomials and #math.equation(block: false, alt: "q not equal to 0 .")[$q ≠ 0 .$] ] Remember, division by 0 is undefined. Here are some examples of rational expressions: #math.equation(block: true, alt: "minus the fraction 13 over 42 the fraction 7 y over 8 z the fraction 5 x plus 2 over x squared minus 7 the fraction 4 x squared plus 3 x minus 1 over 2 x minus 8")[$− frac(13, 42) #h(4em) frac(7 y, 8 z) #h(4em) frac(5 x + 2, x^(2) − 7) #h(4em) frac(4 x^(2) + 3 x − 1, 2 x − 8)$]Notice that the first rational expression listed above, #math.equation(block: false, alt: "minus the fraction 13 over 42 ,")[$− frac(13, 42) ,$] 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 perform the same operations with rational expressions that we do with fractions. We will simplify, add, subtract, multiply, divide, and use them in applications. === Determine the Values for Which a Rational Expression is Undefined 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. If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator. 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 in the set of reals, if possible. ] #examplebox("Example 1")[][ Determine the values for which the rational expression is undefined: ⓐ #math.equation(block: false, alt: "the fraction 9 y over x")[$frac(9 y, x)$] ⓑ #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 9 y over x")[$frac(9 y, x)$]]), [Set the denominator equal to zero. Solve for the variable.], [#math.equation(block: false, alt: "x equals 0")[$x = 0$]], [], [#math.equation(block: false, alt: "the fraction 9 y over x is undefined for x equals 0 .")[$frac(9 y, x) #h(0.2em) "is undefined for" #h(0.2em) x = 0 .$]], )) ⓑ #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")[$frac(4 b − 3, 2 b + 5)$]]), [Set the denominator equal to zero. Solve 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")[$2 b + 5 & = & 0 \ 2 b & = & −5 \ b & = & − frac(5, 2)$]], [], [#math.equation(block: false, alt: "the fraction 4 b minus 3 over 2 b plus 5")[$frac(4 b − 3, 2 b + 5)$] is undefined for #math.equation(block: false, alt: "b equals minus the fraction 5 over 2 .")[$b = − frac(5, 2) .$]], )) ⓒ #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")[$#h(4em) frac(x + 4, x^(2) + 5 x + 6)$]]), [Set the denominator equal to zero. Solve 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")[$x^(2) + 5 x + 6 & = & 0 \ ( x + 2 ) ( x + 3 ) & = & 0 \ x + 2 = 0 #h(0.2em) "or" #h(0.2em) x + 3 & = & 0 \ x = −2 #h(0.2em) "or" #h(0.2em) x & = & −3$]], [], [#math.equation(block: false, alt: "the fraction x plus 4 over x squared plus 5 x plus 6")[$frac(x + 4, x^(2) + 5 x + 6)$] is undefined for #math.equation(block: false, alt: "x equals −2 or x equals −3 .")[$x = −2 #h(0.2em) "or" #h(0.2em) x = −3 .$]], )) Saying that the rational expression #math.equation(block: false, alt: "the fraction x plus 4 over x squared plus 5 x plus 6")[$frac(x + 4, x^(2) + 5 x + 6)$] is undefined for #math.equation(block: false, alt: "x equals −2 or x equals −3")[$x = −2 #h(0.2em) "or" #h(0.2em) x = −3$] is similar to writing the phrase “void where prohibited” in contest rules. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine the values for which the rational expression is undefined: ⓐ #math.equation(block: false, alt: "the fraction 3 y over x")[$frac(3 y, x)$] ⓑ #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")[$n = − frac(1, 3)$] ⓒ #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 values for which the 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")[$y = − frac(2, 3)$] ⓒ #math.equation(block: false, alt: "m equals 2 , m equals −3")[$m = 2 , m = −3$] ] ] === Evaluate Rational Expressions To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book. #examplebox("Example 2")[][ Evaluate #math.equation(block: false, alt: "the fraction 2 x plus 3 over 3 x minus 5")[$frac(2 x + 3, 3 x − 5)$] for each value: ⓐ #math.equation(block: false, alt: "x equals 0")[$x = 0$] ⓑ #math.equation(block: false, alt: "x equals 2")[$x = 2$] ⓒ #math.equation(block: false, alt: "x equals −3")[$x = −3$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression displaying the fraction (2x + 3) over (3x - 5), where 2x + 3 is the numerator and 3x - 5 is the denominator.], alt: "A mathematical expression displaying the fraction (2x + 3) over (3x - 5), where 2x + 3 is the numerator and 3x - 5 is the denominator.", caption: none)]), [#figure(figph[Substitute 0 for x.], alt: "Substitute 0 for x.", caption: none)], [#figure(figph[A mathematical fraction showing the substitution of 0 for a variable in both the numerator and denominator, with the zeroes highlighted in red: (2(0)+3) / (3(0)-5).], alt: "A mathematical fraction showing the substitution of 0 for a variable in both the numerator and denominator, with the zeroes highlighted in red: (2(0)+3) / (3(0)-5).", caption: none)], [Simplify.], [#figure(figph[The image displays the fraction -3/5, representing a negative value where three is divided by five.], alt: "The image displays the fraction -3/5, representing a negative value where three is divided by five.", caption: none)], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical fraction with the expression 2x + 3 in the numerator and 3x - 5 in the denominator, presented in a clear, standard algebraic format.], alt: "A mathematical fraction with the expression 2x + 3 in the numerator and 3x - 5 in the denominator, presented in a clear, standard algebraic format.", caption: none)]), [#figure(figph[The image shows the text 'Substitute 2 for x.' in a gray font, with the number '2' highlighted in a reddish-orange color.], alt: "The image shows the text 'Substitute 2 for x.' in a gray font, with the number '2' highlighted in a reddish-orange color.", caption: none)], [#figure(figph[A mathematical expression displaying a fraction where the variable in the numerator 2(x) + 3 and denominator 3(x) - 5 has been substituted with the number 2, highlighted in red. The expression is 2(2) + 3 / 3(2) - 5.], alt: "A mathematical expression displaying a fraction where the variable in the numerator 2(x) + 3 and denominator 3(x) - 5 has been substituted with the number 2, highlighted in red. The expression is 2(2) + 3 / 3(2) - 5.", caption: none)], [Simplify.], [#figure(figph[A mathematical fraction showing the expression (4+3) in the numerator and (6-5) in the denominator.], alt: "A mathematical fraction showing the expression (4+3) in the numerator and (6-5) in the denominator.", caption: none)], [], [#figure(figph[A mathematical fraction displays the number 7 over the number 1, representing the value of seven divided by one. The numeral 7 is positioned as the numerator above a horizontal line, with the numeral 1 as the denominator below it.], alt: "A mathematical fraction displays the number 7 over the number 1, representing the value of seven divided by one. The numeral 7 is positioned as the numerator above a horizontal line, with the numeral 1 as the denominator below it.", caption: none)], [], [#figure(figph[The number 7, in a dark gray font, is depicted against a clean white background. The number is centrally located and clearly visible, with no other elements or distractions present in the image.], alt: "The number 7, in a dark gray font, is depicted against a clean white background. The number is centrally located and clearly visible, with no other elements or distractions present in the image.", caption: none)], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression displaying a fraction: the numerator is 2x + 3, and the denominator is 3x - 5.], alt: "A mathematical expression displaying a fraction: the numerator is 2x + 3, and the denominator is 3x - 5.", caption: none)]), [#figure(figph[The text reads 'Substitute -3 for x.' on a white background, with the number -3 highlighted in red.], alt: "The text reads 'Substitute -3 for x.' on a white background, with the number -3 highlighted in red.", caption: none)], [#figure(figph[A mathematical fraction is shown, with 2(-3) + 3 in the numerator and 3(-3) - 5 in the denominator. The number -3 is highlighted in red in both parts of the expression.], alt: "A mathematical fraction is shown, with 2(-3) + 3 in the numerator and 3(-3) - 5 in the denominator. The number -3 is highlighted in red in both parts of the expression.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing a fraction with numerator '-6 + 3' and denominator '-9 - 5'.], alt: "A mathematical expression showing a fraction with numerator '-6 + 3' and denominator '-9 - 5'.", caption: none)], [], [#figure(figph[The mathematical expression shows the fraction -3 divided by -14.], alt: "The mathematical expression shows the fraction -3 divided by -14.", caption: none)], [], [#figure(figph[A fraction shows 3 over 14, represented as 3/14, centered on a white background.], alt: "A fraction shows 3 over 14, represented as 3/14, centered on a white background.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "the fraction y plus 1 over 2 y minus 3")[$frac(y + 1, 2 y − 3)$] for each value: ⓐ #math.equation(block: false, alt: "y equals 1")[$y = 1$] ⓑ #math.equation(block: false, alt: "y equals −3")[$y = −3$] ⓒ #math.equation(block: false, alt: "y equals 0")[$y = 0$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−2")[$−2$] ⓑ #math.equation(block: false, alt: "the fraction 2 over 9")[$frac(2, 9)$] ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "the fraction 5 x minus 1 over 2 x plus 1")[$frac(5 x − 1, 2 x + 1)$] for each value: ⓐ #math.equation(block: false, alt: "x equals 1")[$x = 1$] ⓑ #math.equation(block: false, alt: "x equals −1")[$x = −1$] ⓒ #math.equation(block: false, alt: "x equals 0")[$x = 0$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 4 over 3")[$frac(4, 3)$] ⓑ #math.equation(block: false, alt: "6")[$6$] ⓒ #math.equation(block: false, alt: "−1")[$−1$] ] ] #examplebox("Example 3")[][ Evaluate #math.equation(block: false, alt: "the fraction x squared plus 8 x plus 7 over x squared minus 4")[$frac(x^(2) + 8 x + 7, x^(2) − 4)$] for each value: ⓐ #math.equation(block: false, alt: "x equals 0")[$x = 0$] ⓑ #math.equation(block: false, alt: "x equals 2")[$x = 2$] ⓒ #math.equation(block: false, alt: "x equals −1")[$x = −1$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression showing the fraction (x^2 + 8x + 7) divided by (x^2 - 4). The numerator is a quadratic trinomial, and the denominator is a difference of squares.], alt: "A mathematical expression showing the fraction (x^2 + 8x + 7) divided by (x^2 - 4). The numerator is a quadratic trinomial, and the denominator is a difference of squares.", caption: none)]), [#figure(figph[Substitute 0 for x.], alt: "Substitute 0 for x.", caption: none)], [#figure(figph[The evaluation of an algebraic fraction where the variable is replaced by zero, represented as ((0)^2 + 8(0) + 7) / ((0)^2 - 4).], alt: "The evaluation of an algebraic fraction where the variable is replaced by zero, represented as ((0)^2 + 8(0) + 7) / ((0)^2 - 4).", caption: none)], [Simplify.], [#figure(figph[A white background displays the mathematical fraction seven over negative four, representing the value -7/4.], alt: "A white background displays the mathematical fraction seven over negative four, representing the value -7/4.", caption: none)], [], [#figure(figph[The mathematical expression displays a negative fraction, written as minus seven over four (-7/4).], alt: "The mathematical expression displays a negative fraction, written as minus seven over four (-7/4).", caption: none)], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, [], [#figure(figph[A mathematical expression showing the fraction (x^2 + 8x + 7) divided by (x^2 - 4). The numerator is a quadratic trinomial, and the denominator is a difference of squares.], alt: "A mathematical expression showing the fraction (x^2 + 8x + 7) divided by (x^2 - 4). The numerator is a quadratic trinomial, and the denominator is a difference of squares.", caption: none)], [#figure(figph[Substitute 2 for x.], alt: "Substitute 2 for x.", caption: none)], [#figure(figph[A mathematical fraction with (2)^2 + 8(2) + 7 in the numerator and (2)^2 - 4 in the denominator, where the number 2 is highlighted in red.], alt: "A mathematical fraction with (2)^2 + 8(2) + 7 in the numerator and (2)^2 - 4 in the denominator, where the number 2 is highlighted in red.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing the fraction (4 + 16 + 7) / (4 - 4), which results in division by zero, rendering the expression undefined.], alt: "A mathematical expression showing the fraction (4 + 16 + 7) / (4 - 4), which results in division by zero, rendering the expression undefined.", caption: none)], [], [#figure(figph[The mathematical expression 27/0, which is undefined.], alt: "The mathematical expression 27/0, which is undefined.", caption: none)], [This rational expression is undefined for #emph[x] = 2.], [], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression showing a fraction. The numerator is x squared plus 8x plus 7. The denominator is x squared minus 4.], alt: "A mathematical expression showing a fraction. The numerator is x squared plus 8x plus 7. The denominator is x squared minus 4.", caption: none)]), [#figure(figph[The image displays the text 'Substitute -1 for x.', with '-1' highlighted in red.], alt: "The image displays the text 'Substitute -1 for x.', with '-1' highlighted in red.", caption: none)], [#figure(figph[A mathematical fraction showing the evaluation of (x^2 + 8x + 7) / (x^2 - 4) with x = -1, where the -1 is highlighted in red.], alt: "A mathematical fraction showing the evaluation of (x^2 + 8x + 7) / (x^2 - 4) with x = -1, where the -1 is highlighted in red.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing a fraction. The numerator is 1 - 8 + 7 and the denominator is 1 - 4.], alt: "A mathematical expression showing a fraction. The numerator is 1 - 8 + 7 and the denominator is 1 - 4.", caption: none)], [], [#figure(figph[A mathematical expression showing the fraction with numerator -7 + 7 and denominator -3.], alt: "A mathematical expression showing the fraction with numerator -7 + 7 and denominator -3.", caption: none)], [], [#figure(figph[A mathematical expression displaying the fraction 0 over -3, which simplifies to 0. This image shows a fundamental concept in division where zero divided by any non-zero number is zero.], alt: "A mathematical expression displaying the fraction 0 over -3, which simplifies to 0. This image shows a fundamental concept in division where zero divided by any non-zero number is zero.", caption: none)], [], [#figure(figph[A close-up view of the number 0, rendered in a sans-serif font, centered on a plain white background.], alt: "A close-up view of the number 0, rendered in a sans-serif font, centered on a plain white background.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "the fraction x squared plus 1 over x squared minus 3 x plus 2")[$frac(x^(2) + 1, x^(2) − 3 x + 2)$] for each value: ⓐ #math.equation(block: false, alt: "x equals 0")[$x = 0$] ⓑ #math.equation(block: false, alt: "x equals −1")[$x = −1$] ⓒ #math.equation(block: false, alt: "x equals 3")[$x = 3$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] ⓒ 5 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "the fraction x squared plus x minus 6 over x squared minus 9")[$frac(x^(2) + x − 6, x^(2) − 9)$] for each value: ⓐ #math.equation(block: false, alt: "x equals 0")[$x = 0$] ⓑ #math.equation(block: false, alt: "x equals −2")[$x = −2$] ⓒ #math.equation(block: false, alt: "x equals 1")[$x = 1$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] ⓑ #math.equation(block: false, alt: "the fraction 4 over 5")[$frac(4, 5)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] ] ] Remember that a fraction is simplified when it has no common factors, other than 1, in its numerator and denominator. When we evaluate a rational expression, we make sure to simplify the resulting fraction. #examplebox("Example 4")[][ Evaluate #math.equation(block: false, alt: "the fraction a squared plus 2 a b plus b squared over 3 a b squared")[$frac(a^(2) + 2 a b + b^(2), 3 a b^(2))$] for each value: ⓐ #math.equation(block: false, alt: "a equals 1 , b equals 2")[$a = 1 , b = 2$] ⓑ #math.equation(block: false, alt: "a equals −2 , b equals −1")[$a = −2 , b = −1$] ⓒ #math.equation(block: false, alt: "a equals the fraction 1 over 3 , b equals 0")[$a = frac(1, 3) , b = 0$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction a squared plus 2 a b plus b squared over 3 a b squared")[$frac(a^(2) + 2 a b + b^(2), 3 a b^(2))$] when #math.equation(block: false, alt: "a equals 1 , b equals 2")[$a = 1 , b = 2$].]), [#figure(figph[The image displays text in a serif font that reads 'Substitute 1 for a and 2 for b.' The numbers '1' and '2' are highlighted in red and light blue, respectively.], alt: "The image displays text in a serif font that reads 'Substitute 1 for a and 2 for b.' The numbers '1' and '2' are highlighted in red and light blue, respectively.", caption: none)], [#figure(figph[A mathematical fraction with a numerator expanded as (1)^2 + 2(1)(2) + (2)^2 and a denominator as 3(1)(2)^2, featuring red and blue colored numbers.], alt: "A mathematical fraction with a numerator expanded as (1)^2 + 2(1)(2) + (2)^2 and a denominator as 3(1)(2)^2, featuring red and blue colored numbers.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing the fraction (1 + 4 + 4) over 3(4).], alt: "A mathematical expression showing the fraction (1 + 4 + 4) over 3(4).", caption: none)], [], [#figure(figph[The fraction 9/12 is displayed on a white background, with a horizontal line separating the numerator '9' from the denominator '12'.], alt: "The fraction 9/12 is displayed on a white background, with a horizontal line separating the numerator '9' from the denominator '12'.", caption: none)], [], [#figure(figph[The image displays the fraction '3/4' centered on a plain white background. The numerator '3' is positioned above a horizontal fraction bar, with the denominator '4' directly below the bar. The text is black and clear.], alt: "The image displays the fraction '3/4' centered on a plain white background. The numerator '3' is positioned above a horizontal fraction bar, with the denominator '4' directly below the bar. The text is black and clear.", caption: none)], )) #linebreak() ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction a squared plus 2 a b plus b squared over 3 a b squared")[$frac(a^(2) + 2 a b + b^(2), 3 a b^(2))$] when #math.equation(block: false, alt: "a equals −2 , b equals −1")[$a = −2 , b = −1$].]), [#figure(figph[The text instruction 'Substitute -2 for a and -1 for b.' is displayed, with the value -2 highlighted in red and -1 in blue.], alt: "The text instruction 'Substitute -2 for a and -1 for b.' is displayed, with the value -2 highlighted in red and -1 in blue.", caption: none)], [#figure(figph[A mathematical expression featuring a numerator with the sum of squares and a product term, and a denominator with a product of three terms. The numbers -2 and -1 are highlighted in red and light blue respectively.], alt: "A mathematical expression featuring a numerator with the sum of squares and a product term, and a denominator with a product of three terms. The numbers -2 and -1 are highlighted in red and light blue respectively.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing the fraction (4 + 4 + 1) divided by -6, presented in a clean, white background.], alt: "A mathematical expression showing the fraction (4 + 4 + 1) divided by -6, presented in a clean, white background.", caption: none)], [], [#figure(figph[A mathematical expression displaying the fraction -9/6.], alt: "A mathematical expression displaying the fraction -9/6.", caption: none)], [], [#figure(figph[A mathematical expression displaying the negative fraction -3/2, set against a plain white background.], alt: "A mathematical expression displaying the negative fraction -3/2, set against a plain white background.", caption: none)], )) #linebreak() ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction a squared plus 2 a b plus b squared over 3 a b squared")[$frac(a^(2) + 2 a b + b^(2), 3 a b^(2))$] when #math.equation(block: false, alt: "a equals the fraction 1 over 3 , b equals 0")[$a = frac(1, 3) , b = 0$].]), [#figure(figph[The text reads: 'Substitute 1/3 for a and 0 for b.'], alt: "The text reads: 'Substitute 1/3 for a and 0 for b.'", caption: none)], [#figure(figph[A mathematical expression showing a fraction. The numerator is (1/3)^2 + 2(1/3)(0) + (0)^2, and the denominator is 3(1/3)(0)^2. The number 1/3 is in red, and 0 is in blue.], alt: "A mathematical expression showing a fraction. The numerator is (1/3)^2 + 2(1/3)(0) + (0)^2, and the denominator is 3(1/3)(0)^2. The number 1/3 is in red, and 0 is in blue.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing a fraction 1/9 plus two zeros in the numerator, all divided by zero, resulting in an undefined expression due to division by zero.], alt: "A mathematical expression showing a fraction 1/9 plus two zeros in the numerator, all divided by zero, resulting in an undefined expression due to division by zero.", caption: none)], [], [#figure(figph[A mathematical expression showing the fraction 1 over 9, which is then divided by 0, resulting in an undefined or impossible mathematical operation, commonly known as division by zero.], alt: "A mathematical expression showing the fraction 1 over 9, which is then divided by 0, resulting in an undefined or impossible mathematical operation, commonly known as division by zero.", caption: none)], [], [The expression is undefined.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "the fraction 2 a cubed b over a squared plus 2 a b plus b squared")[$frac(2 a^(3) b, a^(2) + 2 a b + b^(2))$] for each value: ⓐ #math.equation(block: false, alt: "a equals −1 , b equals 2")[$a = −1 , b = 2$] ⓑ #math.equation(block: false, alt: "a equals 0 , b equals −1")[$a = 0 , b = −1$] ⓒ #math.equation(block: false, alt: "a equals 1 , b equals the fraction 1 over 2")[$a = 1 , b = frac(1, 2)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−4")[$−4$] ⓑ #math.equation(block: false, alt: "0")[$0$] ⓒ #math.equation(block: false, alt: "the fraction 4 over 9")[$frac(4, 9)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "the fraction a squared minus b squared over 8 a b cubed")[$frac(a^(2) − b^(2), 8 a b^(3))$] for each value: ⓐ #math.equation(block: false, alt: "a equals 1 , b equals −1")[$a = 1 , b = −1$] ⓑ #math.equation(block: false, alt: "a equals the fraction 1 over 2 , b equals −1")[$a = frac(1, 2) , b = −1$] ⓒ #math.equation(block: false, alt: "a equals −2 , b equals 1")[$a = −2 , b = 1$] #solutionbox[ ⓐ #math.equation(block: false, alt: "0")[$0$] ⓑ #math.equation(block: false, alt: "the fraction 3 over 16")[$frac(3, 16)$] ⓒ #math.equation(block: false, alt: "minus the fraction 3 over 16")[$− frac(3, 16)$] ] ] === Simplify Rational Expressions Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is #emph[simplified] if it 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: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] is simplified because there are no common factors of 2 and 3. - #math.equation(block: false, alt: "the fraction 2 x over 3 x")[$frac(2 x, 3 x)$] is not simplified because #emph[x] is a common factor of 2#emph[x] and 3#emph[x]. We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify #strong[rational expression]s. #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")[$b ≠ 0 , c ≠ 0$], 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")[$frac(a · c, b · c) = frac(a, b)$]. ] 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. Every time we write a rational expression, we should make a similar 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. Let’s start by reviewing how we simplify numerical fractions. #examplebox("Example 5")[][ Simplify: #math.equation(block: false, alt: "minus the fraction 36 over 63 .")[$− frac(36, 63) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the fraction -36/63, represented with a horizontal line separating the numerator -36 and the denominator 63.], alt: "The image displays the fraction -36/63, represented with a horizontal line separating the numerator -36 and the denominator 63.", caption: none)]), [Rewrite the numerator and denominator showing the common factors.], [#figure(figph[A mathematical fraction displaying -(4\*9)/(7\*9), with the common multiplier '9' highlighted in red, indicating its cancellation for simplification to -4/7.], alt: "A mathematical fraction displaying -(4*9)/(7*9), with the common multiplier '9' highlighted in red, indicating its cancellation for simplification to -4/7.", caption: none)], [Simplify using the Equivalent Fractions Property.], [#figure(figph[The negative fraction -4/7, showing a minus sign preceding the fraction bar with 4 as the numerator and 7 as the denominator.], alt: "The negative fraction -4/7, showing a minus sign preceding the fraction bar with 4 as the numerator and 7 as the denominator.", caption: none)], )) Notice that the fraction #math.equation(block: false, alt: "minus the fraction 4 over 7")[$− frac(4, 7)$] is simplified because there are no more common factors. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "minus the fraction 45 over 81 .")[$− frac(45, 81) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 9 .")[$− frac(5, 9) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "minus the fraction 42 over 54 .")[$− frac(42, 54) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 7 over 9")[$− frac(7, 9)$] ] ] 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")[$x ≠ 0$] and #math.equation(block: false, alt: "y not equal to 0")[$y ≠ 0$]. #examplebox("Example 6")[][ Simplify: #math.equation(block: false, alt: "the fraction 3 x y over 18 x squared y squared .")[$frac(3 x y, 18 x^(2) y^(2)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A fraction with 3xy in the numerator and 18x²y² in the denominator. This algebraic expression can be simplified by canceling common terms.], alt: "A fraction with 3xy in the numerator and 18x²y² in the denominator. This algebraic expression can be simplified by canceling common terms.", caption: none)]), [Rewrite the numerator and denominator showing the common factors.], [#figure(figph[A fraction with 1 multiplied by 3xy in the numerator and 6xy multiplied by 3xy in the denominator. The 3xy term is highlighted in red, indicating a common factor for simplification.], alt: "A fraction with 1 multiplied by 3xy in the numerator and 6xy multiplied by 3xy in the denominator. The 3xy term is highlighted in red, indicating a common factor for simplification.", caption: none)], [Simplify using the Equivalent Fractions Property.], [#figure(figph[The image displays the mathematical expression one divided by six xy, written as a fraction with 1 as the numerator and 6xy as the denominator.], alt: "The image displays the mathematical expression one divided by six xy, written as a fraction with 1 as the numerator and 6xy as the denominator.", caption: none)], )) Did you notice that these are the same steps we took when we divided monomials in Polynomials? ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 4 x squared y over 12 x y squared .")[$frac(4 x^(2) y, 12 x y^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x over 3 y")[$frac(x, 3 y)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 16 x squared y over 2 x y squared .")[$frac(16 x^(2) y, 2 x y^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 8 x over y")[$frac(8 x, y)$] ] ] 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[This figure contains three columns. The first column, shows the numerator and denominator in factored form. The numerator has 2 times 3 times 7. The denominator has 3 times 5 times 7. The common factors, 3 and 7 are crossed out. The second row, first column shows what remains after the threes and sevens are crossed out, which is 2 over 5 in fraction form. The last row in the first column reads “We removed the common factors of 3 and 7. They are the factors of the product.” The first row of the middle column shows 3 x and then x minus 9 in parentheses in the numerator. The denominator shows 5 and then x-9 in parentheses. The common factors x minus 9 are crossed out. The second row of the middle column shows what remains after removing the common factors, which is 3 x over 5 in fraction form. The last row in the middle column reads, “We removed the common factor x minus 9. It is a factor of the product.” The first row of the third column shows x plus 5 in the numerator and x in the denominator. The second row says “No common factors” and the third row reads, “While there is an x in both the numerator and the denominator, the x in the numerator is a term of a sum”.], alt: "This figure contains three columns. The first column, shows the numerator and denominator in factored form. The numerator has 2 times 3 times 7. The denominator has 3 times 5 times 7. The common factors, 3 and 7 are crossed out. The second row, first column shows what remains after the threes and sevens are crossed out, which is 2 over 5 in fraction form. The last row in the first column reads “We removed the common factors of 3 and 7. They are the factors of the product.” The first row of the middle column shows 3 x and then x minus 9 in parentheses in the numerator. The denominator shows 5 and then x-9 in parentheses. The common factors x minus 9 are crossed out. The second row of the middle column shows what remains after removing the common factors, which is 3 x over 5 in fraction form. The last row in the middle column reads, “We removed the common factor x minus 9. It is a factor of the product.” The first row of the third column shows x plus 5 in the numerator and x in the denominator. The second row says “No common factors” and the third row reads, “While there is an x in both the numerator and the denominator, the x in the numerator is a term of a sum”.", caption: none) Note that 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 7")[How to Simplify Rational Binomials][ Simplify: #math.equation(block: false, alt: "the fraction 2 x plus 8 over 5 x plus 20 .")[$frac(2 x + 8, 5 x + 20) .$] #solutionbox[ #figure(figph[This figure is a table with three columns and two rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell says “Step 1. Factor the numerator and denominator completely.” The second cell says “Factor 2x plus 8 and 5x minus 20.” The third cell contains 2x plus 8, divided by 5x plus 20. Below this is 2 times x plus 4 divided by 5 times x plus 4.], alt: "This figure is a table with three columns and two rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell says “Step 1. Factor the numerator and denominator completely.” The second cell says “Factor 2x plus 8 and 5x minus 20.” The third cell contains 2x plus 8, divided by 5x plus 20. Below this is 2 times x plus 4 divided by 5 times x plus 4.", caption: none) #figure(figph[In the second row, the first cell says “Step 2. Simplify by dividing out common factors.” The second cell says “Divide out the common factors.” The third cell contains 2 times x plus 4 divided by 5 times x plus 4, where x plus 4 cancels out in the numerator and the denominator. It simplifies to 2 fifths.], alt: "In the second row, the first cell says “Step 2. Simplify by dividing out common factors.” The second cell says “Divide out the common factors.” The third cell contains 2 times x plus 4 divided by 5 times x plus 4, where x plus 4 cancels out in the numerator and the denominator. It simplifies to 2 fifths.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 3 x minus 6 over 2 x minus 4 .")[$frac(3 x − 6, 2 x − 4) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 2")[$frac(3, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 7 y plus 35 over 5 y plus 25 .")[$frac(7 y + 35, 5 y + 25) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 5")[$frac(7, 5)$] ] ] 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 all the common factors! We’ll use the methods we covered in Factoring to factor the polynomials in the numerators and denominators in the following examples. #examplebox("Example 8")[][ 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[ #math.equation(block: true, alt: "the fraction x squared plus 5 x plus 6 over x squared plus 8 x plus 12; Factor the numerator and denominator., the fraction open parenthesis x plus 2 close parenthesis open parenthesis x plus 3 close parenthesis over open parenthesis x plus 2 close parenthesis open parenthesis x plus 6 close parenthesis; Remove the common factor x plus 2 from; the numerator and the denominator., the fraction open parenthesis x plus 2 close parenthesis open parenthesis x plus 3 close parenthesis over open parenthesis x plus 2 close parenthesis open parenthesis x plus 6 close parenthesis; the fraction x plus 3 over x plus 6")[$& & & #h(5em) frac(x^(2) + 5 x + 6, x^(2) + 8 x + 12) \ "Factor the numerator and denominator." & & & #h(5em) frac(( x + 2 ) ( x + 3 ), ( x + 2 ) ( x + 6 )) \ \ \ "Remove the common factor" #h(0.2em) x + 2 #h(0.2em) "from" \ "the numerator and the denominator." & & & #h(5em) frac(cancel(( x + 2 )) ( x + 3 ), cancel(( x + 2 )) ( x + 6 )) \ & & & #h(5em) frac(x + 3, x + 6)$] Can you tell which values of #emph[x] must be excluded in this example? ] ] #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")[$frac(x + 1, x − 1)$] ] ] #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")[$frac(x − 5, x − 1)$] ] ] #examplebox("Example 9")[][ Simplify: #math.equation(block: false, alt: "the fraction y squared plus y minus 42 over y squared minus 36 .")[$frac(y^(2) + y − 42, y^(2) − 36) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction y squared plus y minus 42 over y squared minus 36; Factor the numerator and denominator., the fraction open parenthesis y plus 7 close parenthesis open parenthesis y minus 6 close parenthesis over open parenthesis y plus 6 close parenthesis open parenthesis y minus 6 close parenthesis; Remove the common factor y minus 6 from; the numerator and the denominator., the fraction open parenthesis y plus 7 close parenthesis open parenthesis y minus 6 close parenthesis over open parenthesis y plus 6 close parenthesis open parenthesis y minus 6 close parenthesis; the fraction y plus 7 over y plus 6")[$& & & #h(5em) frac(y^(2) + y − 42, y^(2) − 36) \ "Factor the numerator and denominator." & & & #h(5em) frac(( y + 7 ) ( y − 6 ), ( y + 6 ) ( y − 6 )) \ \ \ "Remove the common factor" #h(0.2em) y − 6 #h(0.2em) "from" \ "the numerator and the denominator." & & & #h(5em) frac(( y + 7 ) cancel(( y − 6 )), ( y + 6 ) cancel(( y − 6 ))) \ & & & #h(5em) frac(y + 7, y + 6)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x squared plus x minus 6 over x squared minus 4 .")[$frac(x^(2) + x − 6, x^(2) − 4) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x plus 3 over x plus 2")[$frac(x + 3, x + 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x squared plus 8 x plus 7 over x squared minus 49 .")[$frac(x^(2) + 8 x + 7, x^(2) − 49) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x plus 1 over x minus 7")[$frac(x + 1, x − 7)$] ] ] #examplebox("Example 10")[][ Simplify: #math.equation(block: false, alt: "the fraction p cubed minus 2 p squared plus 2 p minus 4 over p squared minus 7 p plus 10 .")[$frac(p^(3) − 2 p^(2) + 2 p − 4, p^(2) − 7 p + 10) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction p cubed minus 2 p squared plus 2 p minus 4 over p squared minus 7 p plus 10; Factor the numerator and denominator,; using grouping to factor the numerator., the fraction p squared open parenthesis p minus 2 close parenthesis plus 2 open parenthesis p minus 2 close parenthesis over open parenthesis p minus 5 close parenthesis open parenthesis p minus 2 close parenthesis; the fraction open parenthesis p squared plus 2 close parenthesis open parenthesis p minus 2 close parenthesis over open parenthesis p minus 5 close parenthesis open parenthesis p minus 2 close parenthesis; Remove the common factor of p minus 2; from the numerator and the denominator., the fraction open parenthesis p squared plus 2 close parenthesis open parenthesis p minus 2 close parenthesis over open parenthe")[$& & & #h(5em) frac(p^(3) − 2 p^(2) + 2 p − 4, p^(2) − 7 p + 10) \ \ \ "Factor the numerator and denominator," \ "using grouping to factor the numerator." & & & #h(5em) frac(p^(2) ( p − 2 ) + 2 ( p − 2 ), ( p − 5 ) ( p − 2 )) \ & & & #h(5em) frac(( p^(2) + 2 ) ( p − 2 ), ( p − 5 ) ( p − 2 )) \ \ \ "Remove the common factor of" #h(0.2em) p − 2 \ "from the numerator and the denominator." & & & #h(5em) frac(( p^(2) + 2 ) cancel(( p − 2 )), ( p − 5 ) cancel(( p − 2 ))) \ & & & #h(5em) frac(p^(2) + 2, p − 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction y cubed minus 3 y squared plus y minus 3 over y squared minus y minus 6 .")[$frac(y^(3) − 3 y^(2) + y − 3, y^(2) − y − 6) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction y squared plus 1 over y plus 2")[$frac(y^(2) + 1, y + 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction p cubed minus p squared plus 2 p minus 2 over p squared plus 4 p minus 5 .")[$frac(p^(3) − p^(2) + 2 p − 2, p^(2) + 4 p − 5) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction p squared plus 2 over p plus 5")[$frac(p^(2) + 2, p + 5)$] ] ] #examplebox("Example 11")[][ Simplify: #math.equation(block: false, alt: "the fraction 2 n squared minus 14 n over 4 n squared minus 16 n minus 48 .")[$frac(2 n^(2) − 14 n, 4 n^(2) − 16 n − 48) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 n squared minus 14 n over 4 n squared minus 16 n minus 48; Factor the numerator and denominator,; first factoring out the GCF., the fraction 2 n open parenthesis n minus 7 close parenthesis over 4 open parenthesis n squared minus 4 n minus 12 close parenthesis; the fraction 2 n open parenthesis n minus 7 close parenthesis over 4 open parenthesis n minus 6 close parenthesis open parenthesis n plus 2 close parenthesis; Remove the common factor, 2., the fraction 2 n open parenthesis n minus 7 close parenthesis over 2 times 2 open parenthesis n minus 6 close parenthesis open parenthesis n plus 2 close parenthesis; the fraction n open parenthesis n minus 7 close parenthesis over 2 open parenthesis n minus 6 close parenthesis open parenthesis n plus 2 close parenthesis")[$& & & #h(5em) frac(2 n^(2) − 14 n, 4 n^(2) − 16 n − 48) \ \ \ "Factor the numerator and denominator," \ "first factoring out the GCF." & & & #h(5em) frac(2 n ( n − 7 ), 4 ( n^(2) − 4 n − 12 )) \ & & & #h(5em) frac(2 n ( n − 7 ), 4 ( n − 6 ) ( n + 2 )) \ \ \ "Remove the common factor, 2." & & & #h(5em) frac(cancel(2) n ( n − 7 ), cancel(2) · 2 ( n − 6 ) ( n + 2 )) \ & & & #h(5em) frac(n ( n − 7 ), 2 ( n − 6 ) ( n + 2 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 2 n squared minus 10 n over 4 n squared minus 16 n minus 20 .")[$frac(2 n^(2) − 10 n, 4 n^(2) − 16 n − 20) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction n over 2 open parenthesis n plus 1 close parenthesis")[$frac(n, 2 ( n + 1 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 4 x squared minus 16 x over 8 x squared minus 16 x minus 64 .")[$frac(4 x^(2) − 16 x, 8 x^(2) − 16 x − 64) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x over 2 open parenthesis x plus 2 close parenthesis")[$frac(x, 2 ( x + 2 ))$] ] ] #examplebox("Example 12")[][ Simplify: #math.equation(block: false, alt: "the fraction 3 b squared minus 12 b plus 12 over 6 b squared minus 24 .")[$frac(3 b^(2) − 12 b + 12, 6 b^(2) − 24) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 b squared minus 12 b plus 12 over 6 b squared minus 24; Factor the numerator and denominator,; first factoring out the GCF., the fraction 3 open parenthesis b squared minus 4 b plus 4 close parenthesis over 6 open parenthesis b squared minus 4 close parenthesis; the fraction 3 open parenthesis b minus 2 close parenthesis open parenthesis b minus 2 close parenthesis over 6 open parenthesis b plus 2 close parenthesis open parenthesis b minus 2 close parenthesis; Remove the common factors of b minus 2 and 3 ., the fraction 3 open parenthesis b minus 2 close parenthesis open parenthesis b minus 2 close parenthesis over 3 times 2 open parenthesis b plus 2 close parenthesis open parenthesis b minus 2 close parenthesis; the fraction b minus 2 over 2 open parenthesis b plus 2 close")[$& & & #h(5em) frac(3 b^(2) − 12 b + 12, 6 b^(2) − 24) \ \ \ "Factor the numerator and denominator," \ "first factoring out the GCF." & & & #h(5em) frac(3 ( b^(2) − 4 b + 4 ), 6 ( b^(2) − 4 )) \ & & & #h(5em) frac(3 ( b − 2 ) ( b − 2 ), 6 ( b + 2 ) ( b − 2 )) \ \ \ "Remove the common factors of" #h(0.2em) b − 2 #h(0.2em) "and" #h(0.2em) 3 . & & & #h(5em) frac(cancel(3) ( b − 2 ) cancel(( b − 2 )), cancel(3) · 2 ( b + 2 ) cancel(( b − 2 ))) \ & & & #h(5em) frac(b − 2, 2 ( b + 2 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 2 x squared minus 12 x plus 18 over 3 x squared minus 27 .")[$frac(2 x^(2) − 12 x + 18, 3 x^(2) − 27) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 open parenthesis x minus 3 close parenthesis over 3 open parenthesis x plus 3 close parenthesis")[$frac(2 ( x − 3 ), 3 ( x + 3 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 5 y squared minus 30 y plus 25 over 2 y squared minus 50 .")[$frac(5 y^(2) − 30 y + 25, 2 y^(2) − 50) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 open parenthesis y minus 1 close parenthesis over 2 open parenthesis y plus 5 close parenthesis")[$frac(5 ( y − 1 ), 2 ( y + 5 ))$] ] ] #examplebox("Example 13")[][ Simplify: #math.equation(block: false, alt: "the fraction m cubed plus 8 over m squared minus 4 .")[$frac(m^(3) + 8, m^(2) − 4) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction m cubed plus 8 over m squared minus 4; Factor the numerator and denominator,; using the formulas for sum of cubes and; difference of squares., the fraction open parenthesis m plus 2 close parenthesis open parenthesis m squared minus 2 m plus 4 close parenthesis over open parenthesis m plus 2 close parenthesis open parenthesis m minus 2 close parenthesis; Remove the common factor of m plus 2 ., the fraction open parenthesis m plus 2 close parenthesis open parenthesis m squared minus 2 m plus 4 close parenthesis over open parenthesis m plus 2 close parenthesis open parenthesis m minus 2 close parenthesis; the fraction m squared minus 2 m plus 4 over m minus 2")[$& & & #h(5em) frac(m^(3) + 8, m^(2) − 4) \ "Factor the numerator and denominator," \ "using the formulas for sum of cubes and" \ "difference of squares." & & & #h(5em) frac(( m + 2 ) ( m^(2) − 2 m + 4 ), ( m + 2 ) ( m − 2 )) \ "Remove the common factor of" #h(0.2em) m + 2 . & & & #h(5em) frac(cancel(( m + 2 )) ( m^(2) − 2 m + 4 ), cancel(( m + 2 )) ( m − 2 )) \ & & & #h(5em) frac(m^(2) − 2 m + 4, m − 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction p cubed minus 64 over p squared minus 16 .")[$frac(p^(3) − 64, p^(2) − 16) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction p squared plus 4 p plus 16 over p plus 4")[$frac(p^(2) + 4 p + 16, p + 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x cubed plus 8 over x squared minus 4 .")[$frac(x^(3) + 8, x^(2) − 4) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x squared minus 2 x plus 4 over x minus 2")[$frac(x^(2) − 2 x + 4, x − 2)$] ] ] === Simplify Rational Expressions with Opposite Factors Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say #math.equation(block: false, alt: "the fraction 7 over −7")[$frac(7, −7)$]. We know this fraction simplifies to #math.equation(block: false, alt: "−1")[$−1$]. We also recognize that the numerator and denominator are opposites. In Foundations, we introduced opposite notation: the opposite of #math.equation(block: false, alt: "a")[$a$] is #math.equation(block: false, alt: "− a")[$"−" a$]. We remember, too, that #math.equation(block: false, alt: "− a equals −1 times a")[$"−" a = −1 · a$]. We simplify the fraction #math.equation(block: false, alt: "the fraction a over − a")[$frac(a, "−" a)$], whose numerator and denominator are opposites, in this way: #math.equation(block: true, alt: "the fraction a over − a; We could rewrite this., the fraction 1 times a over −1 times a; Remove the common factors., the fraction 1 over −1; Simplify., −1")[$& & & #h(5em) frac(a, "−" a) \ "We could rewrite this." & & & #h(5em) frac(1 · a, −1 · a) \ "Remove the common factors." & & & #h(5em) frac(1, −1) \ "Simplify." & & & #h(5em) −1$] So, in the same way, we can simplify the fraction #math.equation(block: false, alt: "the fraction x minus 3 over − open parenthesis x minus 3 close parenthesis")[$frac(x − 3, "−" ( x − 3 ))$]: #math.equation(block: true, alt: "We could rewrite this., the fraction 1 times open parenthesis x minus 3 close parenthesis over −1 times open parenthesis x minus 3 close parenthesis; Remove the common factors., the fraction 1 over −1; Simplify., −1")[$"We could rewrite this." & & & #h(5em) frac(1 · ( x − 3 ), −1 · ( x − 3 )) \ "Remove the common factors." & & & #h(5em) frac(1, −1) \ "Simplify." & & & #h(5em) −1$] But the opposite of #math.equation(block: false, alt: "x minus 3")[$x − 3$] could be written differently: #math.equation(block: true, alt: "− open parenthesis x minus 3 close parenthesis; Distribute., − x plus 3; Rewrite., 3 minus x")[$& & & #h(12.5em) "−" ( x − 3 ) \ "Distribute." & & & #h(12.5em) "−" x + 3 \ "Rewrite." & & & #h(12.5em) 3 − x$] This means the fraction #math.equation(block: false, alt: "the fraction x minus 3 over 3 minus x")[$frac(x − 3, 3 − x)$] simplifies to #math.equation(block: false, alt: "−1")[$−1$]. 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")[$frac(a − b, b − a) = −1 & & & & & a ≠ b$]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. #examplebox("Example 14")[][ Simplify: #math.equation(block: false, alt: "the fraction x minus 8 over 8 minus x .")[$frac(x − 8, 8 − x) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x minus 8 over 8 minus x; Recognize that x minus 8 and 8 minus x are opposites., −1")[$& & & frac(x − 8, 8 − x) \ "Recognize that" #h(0.2em) x − 8 #h(0.2em) "and" #h(0.2em) 8 − x #h(0.2em) "are opposites." & & & −1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction y minus 2 over 2 minus y .")[$frac(y − 2, 2 − y) .$] #solutionbox[ #math.equation(block: true, alt: "−1")[$−1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction n minus 9 over 9 minus n .")[$frac(n − 9, 9 − n) .$] #solutionbox[ #math.equation(block: true, alt: "−1")[$−1$] ] ] Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely. #examplebox("Example 15")[][ Simplify: #math.equation(block: false, alt: "the fraction 14 minus 2 x over x squared minus 49 .")[$frac(14 − 2 x, x^(2) − 49) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression displaying the fraction (14 - 2x) over (x^2 - 49).], alt: "A mathematical expression displaying the fraction (14 - 2x) over (x^2 - 49).", caption: none)]), [Factor the numerator and denominator.], [#figure(figph[A mathematical expression showing the fraction 2(7-x) over (x+7)(x-7).], alt: "A mathematical expression showing the fraction 2(7-x) over (x+7)(x-7).", caption: none)], [Recognize that #math.equation(block: false, alt: "7 minus x and x minus 7 are opposites")[$7 − x #h(0.2em) "and" #h(0.2em) x − 7 #h(0.2em) "are opposites"$].], [#figure(figph[A math expression showing the fraction 2(7-x) over (x+7)(x-7). Diagonal lines on (7-x) in the numerator and (x-7) in the denominator indicate their cancellation. The fraction is multiplied by a red (-1).], alt: "A math expression showing the fraction 2(7-x) over (x+7)(x-7). Diagonal lines on (7-x) in the numerator and (x-7) in the denominator indicate their cancellation. The fraction is multiplied by a red (-1).", caption: none)], [Simplify.], [#figure(figph[A mathematical expression displaying a fraction: negative 2 over the quantity of x plus 7. The fraction is written with a horizontal line separating the numerator 2 from the denominator x+7, with a minus sign to the left of the fraction line.], alt: "A mathematical expression displaying a fraction: negative 2 over the quantity of x plus 7. The fraction is written with a horizontal line separating the numerator 2 from the denominator x+7, with a minus sign to the left of the fraction line.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 10 minus 2 y over y squared minus 25 .")[$frac(10 − 2 y, y^(2) − 25) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over y plus 5 .")[$− frac(2, y + 5) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 3 y minus 27 over 81 minus y squared .")[$frac(3 y − 27, 81 − y^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 9 plus y")[$− frac(3, 9 + y)$] ] ] #examplebox("Example 16")[][ 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 fraction with the numerator x squared minus 4x minus 32, and the denominator 64 minus x squared.], alt: "A fraction with the numerator x squared minus 4x minus 32, and the denominator 64 minus x squared.", caption: none)]), [Factor the numerator and denominator.], [#figure(figph[A mathematical fraction is displayed with (x-8)(x+4) in the numerator and (8-x)(8+x) in the denominator.], alt: "A mathematical fraction is displayed with (x-8)(x+4) in the numerator and (8-x)(8+x) in the denominator.", caption: none)], [Recognize the factors that are opposites.], [#figure(figph[A math expression showing the cancellation of (x-8) and (8-x) terms in a rational function. The factor of -1 is explicitly shown, indicating how (x-8) = -(8-x) is used for simplification.], alt: "A math expression showing the cancellation of (x-8) and (8-x) terms in a rational function. The factor of -1 is explicitly shown, indicating how (x-8) = -(8-x) is used for simplification.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing a negative fraction. The numerator is x plus 4, and the denominator is x plus 8. The entire fraction is negated.], alt: "A mathematical expression showing a negative fraction. The numerator is x plus 4, and the denominator is x plus 8. The entire fraction is negated.", 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")[$− frac(x + 2, x + 1)$] ] ] === Key Concepts - #strong[Determine the Values for Which a Rational Expression is Undefined] + Set the denominator equal to zero. + Solve the equation, if possible. - #strong[Simplified Rational Expression] - A rational expression is considered simplified if there are no common factors in its numerator and denominator. - #strong[Simplify a Rational Expression] + Factor the numerator and denominator completely. + Simplify by dividing out common factors. - #strong[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")[$b − a$]. #linebreak() #math.equation(block: false, alt: "the fraction a minus b over b minus a equals −1, a not equal to 0 , b not equal to 0 , a not equal to b")[$frac(a − b, b − a) = −1 & & & & & a ≠ 0 , b ≠ 0 , "a" ≠ "b"$] ==== Practice Makes Perfect In the following exercises, determine the values for which the rational expression is undefined. ⓐ #math.equation(block: false, alt: "the fraction 2 x over z")[$frac(2 x, z)$] ⓑ #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")[$p = frac(5, 6)$]ⓒ #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[Evaluate Rational Expressions] In the following exercises, evaluate the rational expression for the given values. #math.equation(block: true, alt: "the fraction 2 x over x minus 1")[$frac(2 x, x − 1)$] ⓐ #math.equation(block: false, alt: "x equals 0")[$x = 0$] ⓑ #math.equation(block: false, alt: "x equals 2")[$x = 2$] ⓒ #math.equation(block: false, alt: "x equals −1")[$x = −1$] #solutionbox[ ⓐ #math.equation(block: false, alt: "0")[$0$] ⓑ #math.equation(block: false, alt: "4")[$4$] ⓒ #math.equation(block: false, alt: "1")[$1$] ] #math.equation(block: true, alt: "the fraction 4 y minus 1 over 5 y minus 3")[$frac(4 y − 1, 5 y − 3)$] ⓐ #math.equation(block: false, alt: "y equals 0")[$y = 0$] ⓑ #math.equation(block: false, alt: "y equals 2")[$y = 2$] ⓒ #math.equation(block: false, alt: "y equals −1")[$y = −1$] #math.equation(block: true, alt: "the fraction 2 p plus 3 over p squared plus 1")[$frac(2 p + 3, p^(2) + 1)$] ⓐ #math.equation(block: false, alt: "p equals 0")[$p = 0$] ⓑ #math.equation(block: false, alt: "p equals 1")[$p = 1$] ⓒ #math.equation(block: false, alt: "p equals −2")[$p = −2$] #solutionbox[ ⓐ #math.equation(block: false, alt: "3")[$3$] ⓑ #math.equation(block: false, alt: "the fraction 5 over 2")[$frac(5, 2)$] ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 5")[$− frac(1, 5)$] ] #math.equation(block: true, alt: "the fraction x plus 3 over 2 minus 3 x")[$frac(x + 3, 2 − 3 x)$] ⓐ #math.equation(block: false, alt: "x equals 0")[$x = 0$] ⓑ #math.equation(block: false, alt: "x equals 1")[$x = 1$] ⓒ #math.equation(block: false, alt: "x equals −2")[$x = −2$] #math.equation(block: true, alt: "the fraction y squared plus 5 y plus 6 over y squared minus 1")[$frac(y^(2) + 5 y + 6, y^(2) − 1)$] ⓐ #math.equation(block: false, alt: "y equals 0")[$y = 0$] ⓑ #math.equation(block: false, alt: "y equals 2")[$y = 2$] ⓒ #math.equation(block: false, alt: "y equals −2")[$y = −2$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−6")[$−6$] ⓑ #math.equation(block: false, alt: "the fraction 20 over 3")[$frac(20, 3)$] ⓒ #math.equation(block: false, alt: "0")[$0$] ] #math.equation(block: true, alt: "the fraction z squared plus 3 z minus 10 over z squared minus 1")[$frac(z^(2) + 3 z − 10, z^(2) − 1)$] ⓐ #math.equation(block: false, alt: "z equals 0")[$z = 0$] ⓑ #math.equation(block: false, alt: "z equals 2")[$z = 2$] ⓒ #math.equation(block: false, alt: "z equals −2")[$z = −2$] #math.equation(block: true, alt: "the fraction a squared minus 4 over a squared plus 5 a plus 4")[$frac(a^(2) − 4, a^(2) + 5 a + 4)$] ⓐ #math.equation(block: false, alt: "a equals 0")[$a = 0$] ⓑ #math.equation(block: false, alt: "a equals 1")[$a = 1$] ⓒ #math.equation(block: false, alt: "a equals −2")[$a = −2$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−1")[$−1$] ⓑ #math.equation(block: false, alt: "minus the fraction 3 over 10")[$− frac(3, 10)$] ⓒ #math.equation(block: false, alt: "0")[$0$] ] #math.equation(block: true, alt: "the fraction b squared plus 2 over b squared minus 3 b minus 4")[$frac(b^(2) + 2, b^(2) − 3 b − 4)$] ⓐ #math.equation(block: false, alt: "b equals 0")[$b = 0$] ⓑ #math.equation(block: false, alt: "b equals 2")[$b = 2$] ⓒ #math.equation(block: false, alt: "b equals −2")[$b = −2$] #math.equation(block: true, alt: "the fraction x squared plus 3 x y plus 2 y squared over 2 x cubed y")[$frac(x^(2) + 3 x y + 2 y^(2), 2 x^(3) y)$] + ⓐ #math.equation(block: false, alt: "x equals 1 , y equals −1")[$x = 1 , y = −1$] + ⓑ #math.equation(block: false, alt: "x equals 2 , y equals 1")[$x = 2 , y = 1$] + ⓒ #math.equation(block: false, alt: "x equals −1 , y equals −2")[$x = −1 , y = −2$] #solutionbox[ ⓐ #math.equation(block: false, alt: "0")[$0$] ⓑ #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] ⓒ #math.equation(block: false, alt: "the fraction 15 over 4")[$frac(15, 4)$] ] #math.equation(block: true, alt: "the fraction c squared plus c d minus 2 d squared over c d cubed")[$frac(c^(2) + c d − 2 d^(2), c d^(3))$] + ⓐ #math.equation(block: false, alt: "c equals 2 , d equals −1")[$c = 2 , d = −1$] + ⓑ #math.equation(block: false, alt: "c equals 1 , d equals −1")[$c = 1 , d = −1$] + ⓒ #math.equation(block: false, alt: "c equals −1 , d equals 2")[$c = −1 , d = 2$] #math.equation(block: true, alt: "the fraction m squared minus 4 n squared over 5 m n cubed")[$frac(m^(2) − 4 n^(2), 5 m n^(3))$] + ⓐ #math.equation(block: false, alt: "m equals 2 , n equals 1")[$m = 2 , n = 1$] + ⓑ #math.equation(block: false, alt: "m equals −1 , n equals −1")[$m = −1 , n = −1$] + ⓒ #math.equation(block: false, alt: "m equals 3 , n equals 2")[$m = 3 , n = 2$] #solutionbox[ ⓐ #math.equation(block: false, alt: "0")[$0$] ⓑ #math.equation(block: false, alt: "minus the fraction 3 over 5")[$− frac(3, 5)$] ⓒ #math.equation(block: false, alt: "minus the fraction 7 over 120")[$− frac(7, 120)$] ] #math.equation(block: true, alt: "the fraction 2 s squared t over s squared minus 9 t squared")[$frac(2 s^(2) t, s^(2) − 9 t^(2))$] + ⓐ #math.equation(block: false, alt: "s equals 4 , t equals 1")[$s = 4 , t = 1$] + ⓑ #math.equation(block: false, alt: "s equals −1 , t equals −1")[$s = −1 , t = −1$] + ⓒ #math.equation(block: false, alt: "s equals 0 , t equals 2")[$s = 0 , t = 2$] #strong[Simplify Rational Expressions] In the following exercises, simplify. #math.equation(block: true, alt: "minus the fraction 4 over 52")[$− frac(4, 52)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 13")[$− frac(1, 13)$] ] #math.equation(block: true, alt: "minus the fraction 44 over 55")[$− frac(44, 55)$] #math.equation(block: true, alt: "the fraction 56 over 63")[$frac(56, 63)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 8 over 9")[$frac(8, 9)$] ] #math.equation(block: true, alt: "the fraction 65 over 104")[$frac(65, 104)$] #math.equation(block: true, alt: "the fraction 6 a b squared over 12 a squared b")[$frac(6 a b^(2), 12 a^(2) b)$] #solutionbox[ #math.equation(block: true, alt: "the fraction b over 2 a")[$frac(b, 2 a)$] ] #math.equation(block: true, alt: "the fraction 15 x y over 3 x cubed y cubed")[$frac(15 x y, 3 x^(3) y^(3))$] #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 3 a plus 6 over 4 a plus 8")[$frac(3 a + 6, 4 a + 8)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] #math.equation(block: true, alt: "the fraction 5 b plus 5 over 6 b plus 6")[$frac(5 b + 5, 6 b + 6)$] #math.equation(block: true, alt: "the fraction 3 c minus 9 over 5 c minus 15")[$frac(3 c − 9, 5 c − 15)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 5")[$frac(3, 5)$] ] #math.equation(block: true, alt: "the fraction 4 d plus 8 over 9 d plus 18")[$frac(4 d + 8, 9 d + 18)$] #math.equation(block: true, alt: "the fraction 7 m plus 63 over 5 m plus 45")[$frac(7 m + 63, 5 m + 45)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 5")[$frac(7, 5)$] ] #math.equation(block: true, alt: "the fraction 8 n minus 96 over 3 n minus 36")[$frac(8 n − 96, 3 n − 36)$] #math.equation(block: true, alt: "the fraction 12 p minus 240 over 5 p minus 100")[$frac(12 p − 240, 5 p − 100)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 12 over 5")[$frac(12, 5)$] ] #math.equation(block: true, alt: "the fraction 6 q plus 210 over 5 q plus 175")[$frac(6 q + 210, 5 q + 175)$] #math.equation(block: true, alt: "the fraction a squared minus a minus 12 over a squared minus 8 a plus 16")[$frac(a^(2) − a − 12, a^(2) − 8 a + 16)$] #solutionbox[ #math.equation(block: true, alt: "the fraction a plus 3 over a minus 4")[$frac(a + 3, a − 4)$] ] #math.equation(block: true, alt: "the fraction x squared plus 4 x minus 5 over x squared minus 2 x plus 1")[$frac(x^(2) + 4 x − 5, x^(2) − 2 x + 1)$] #math.equation(block: true, alt: "the fraction y squared plus 3 y minus 4 over y squared minus 6 y plus 5")[$frac(y^(2) + 3 y − 4, y^(2) − 6 y + 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction y plus 4 over y minus 5")[$frac(y + 4, y − 5)$] ] #math.equation(block: true, alt: "the fraction v squared plus 8 v plus 15 over v squared minus v minus 12")[$frac(v^(2) + 8 v + 15, v^(2) − v − 12)$] #math.equation(block: true, alt: "the fraction x squared minus 25 over x squared plus 2 x minus 15")[$frac(x^(2) − 25, x^(2) + 2 x − 15)$] #solutionbox[ #math.equation(block: true, alt: "the fraction x minus 5 over x minus 3")[$frac(x − 5, x − 3)$] ] #math.equation(block: true, alt: "the fraction a squared minus 4 over a squared plus 6 a minus 16")[$frac(a^(2) − 4, a^(2) + 6 a − 16)$] #math.equation(block: true, alt: "the fraction y squared minus 2 y minus 3 over y squared minus 9")[$frac(y^(2) − 2 y − 3, y^(2) − 9)$] #solutionbox[ #math.equation(block: true, alt: "the fraction y plus 1 over y plus 3")[$frac(y + 1, y + 3)$] ] #math.equation(block: true, alt: "the fraction b squared plus 9 b plus 18 over b squared minus 36")[$frac(b^(2) + 9 b + 18, b^(2) − 36)$] #math.equation(block: true, alt: "the fraction y cubed plus y squared plus y plus 1 over y squared plus 2 y plus 1")[$frac(y^(3) + y^(2) + y + 1, y^(2) + 2 y + 1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction y squared plus 1 over y plus 1")[$frac(y^(2) + 1, y + 1)$] ] #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")[$frac(p^(3) + 3 p^(2) + 4 p + 12, p^(2) + p − 6)$] #math.equation(block: true, alt: "the fraction x cubed minus 2 x squared minus 25 x plus 50 over x squared minus 25")[$frac(x^(3) − 2 x^(2) − 25 x + 50, x^(2) − 25)$] #solutionbox[ #math.equation(block: true, alt: "x minus 2")[$x − 2$] ] #math.equation(block: true, alt: "the fraction q cubed plus 3 q squared minus 4 q minus 12 over q squared minus 4")[$frac(q^(3) + 3 q^(2) − 4 q − 12, q^(2) − 4)$] #math.equation(block: true, alt: "the fraction 3 a squared plus 15 a over 6 a squared plus 6 a minus 36")[$frac(3 a^(2) + 15 a, 6 a^(2) + 6 a − 36)$] #solutionbox[ #math.equation(block: true, alt: "the fraction a open parenthesis a plus 5 close parenthesis over 2 open parenthesis a plus 3 close parenthesis open parenthesis a minus 2 close parenthesis")[$frac(a ( a + 5 ), 2 ( a + 3 ) ( a − 2 ))$] ] #math.equation(block: true, alt: "the fraction 8 b squared minus 32 b over 2 b squared minus 6 b minus 80")[$frac(8 b^(2) − 32 b, 2 b^(2) − 6 b − 80)$] #math.equation(block: true, alt: "the fraction −5 c squared minus 10 c over −10 c squared plus 30 c plus 100")[$frac(−5 c^(2) − 10 c, −10 c^(2) + 30 c + 100)$] #solutionbox[ #math.equation(block: true, alt: "the fraction c over 2 open parenthesis c minus 5 close parenthesis")[$frac(c, 2 ( c − 5 ))$] ] #math.equation(block: true, alt: "the fraction 4 d squared minus 24 d over 2 d squared minus 4 d minus 48")[$frac(4 d^(2) − 24 d, 2 d^(2) − 4 d − 48)$] #math.equation(block: true, alt: "the fraction 3 m squared plus 30 m plus 75 over 4 m squared minus 100")[$frac(3 m^(2) + 30 m + 75, 4 m^(2) − 100)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 open parenthesis m plus 5 close parenthesis over 4 open parenthesis m minus 5 close parenthesis")[$frac(3 ( m + 5 ), 4 ( m − 5 ))$] ] #math.equation(block: true, alt: "the fraction 5 n squared plus 30 n plus 45 over 2 n squared minus 18")[$frac(5 n^(2) + 30 n + 45, 2 n^(2) − 18)$] #math.equation(block: true, alt: "the fraction 5 r squared plus 30 r minus 35 over r squared minus 49")[$frac(5 r^(2) + 30 r − 35, r^(2) − 49)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 open parenthesis r minus 1 close parenthesis over r minus 7")[$frac(5 ( r − 1 ), r − 7)$] ] #math.equation(block: true, alt: "the fraction 3 s squared plus 30 s plus 72 over 3 s squared minus 48")[$frac(3 s^(2) + 30 s + 72, 3 s^(2) − 48)$] #math.equation(block: true, alt: "the fraction t cubed minus 27 over t squared minus 9")[$frac(t^(3) − 27, t^(2) − 9)$] #solutionbox[ #math.equation(block: true, alt: "the fraction t squared plus 3 t plus 9 over t plus 3")[$frac(t^(2) + 3 t + 9, t + 3)$] ] #math.equation(block: true, alt: "the fraction v cubed minus 1 over v squared minus 1")[$frac(v^(3) − 1, v^(2) − 1)$] #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")[$frac(v^(3) + 125, v^(2) − 25)$] #strong[Simplify Rational Expressions with Opposite Factors] In the following exercises, simplify each rational expression. #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 b minus 12 over 12 minus b")[$frac(b − 12, 12 − b)$] #math.equation(block: true, alt: "the fraction 11 minus c over c minus 11")[$frac(11 − c, c − 11)$] #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 12 minus 2 x over x squared minus 36")[$frac(12 − 2 x, x^(2) − 36)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over x plus 6")[$− frac(2, x + 6)$] ] #math.equation(block: true, alt: "the fraction 20 minus 5 y over y squared minus 16")[$frac(20 − 5 y, y^(2) − 16)$] #math.equation(block: true, alt: "the fraction 4 v minus 32 over 64 minus v squared")[$frac(4 v − 32, 64 − v^(2))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 over 8 plus v")[$− frac(4, 8 + v)$] ] #math.equation(block: true, alt: "the fraction 7 w minus 21 over 9 minus w squared")[$frac(7 w − 21, 9 − w^(2))$] #math.equation(block: true, alt: "the fraction y squared minus 11 y plus 24 over 9 minus y squared")[$frac(y^(2) − 11 y + 24, 9 − y^(2))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction open parenthesis y minus 8 close parenthesis over 3 plus y")[$− frac(( y − 8 ), 3 + y)$] ] #math.equation(block: true, alt: "the fraction z squared minus 9 z plus 20 over 16 minus z squared")[$frac(z^(2) − 9 z + 20, 16 − z^(2))$] #math.equation(block: true, alt: "the fraction a squared minus 5 a minus 36 over 81 minus a squared")[$frac(a^(2) − 5 a − 36, 81 − a^(2))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction a plus 4 over 9 plus a")[$− frac(a + 4, 9 + a)$] ] #math.equation(block: true, alt: "the fraction b squared plus b minus 42 over 36 minus b squared")[$frac(b^(2) + b − 42, 36 − b^(2))$] ==== Everyday Math #strong[Tax Rates] For the tax year 2015, the amount of tax owed by a single person earning between \$37,450 and \$90,750, can be found by evaluating the formula #math.equation(block: false, alt: "0.25 x minus 4206.25 ,")[$0.25 x − 4206.25 ,$] where #emph[x] is income. The average tax rate for this income can be found by evaluating the formula #math.equation(block: false, alt: "the fraction 0.25 x minus 4206.25 over x .")[$frac(0.25 x − 4206.25, x) .$] What would be the average tax rate for a single person earning \$50,000? #solutionbox[ 16.6% ] #strong[Work] The length of time it takes for two people for perform the same task if they work together can be found by evaluating the formula #math.equation(block: false, alt: "the fraction x y over x plus y .")[$frac(x y, x + y) .$] If Tom can paint the den in #math.equation(block: false, alt: "x equals")[$x =$] 45 minutes and his brother Bobby can paint it in #math.equation(block: false, alt: "y equals")[$y =$] 60 minutes, how many minutes will it take them if they work together? ==== 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, but all should reference setting the denominator function to zero. ] 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 .")[$frac(p^(2) + 4 p − 21, 9 − p^(2)) .$] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This figure shows a table with four columns and five rows. The first row is a header row and each column is labeled. The first column header is labeled “I can…”, the second is labeled “Confidently”, the third is labeled “With some help”, and the fourth is labeled “No—I don’t get it!” In the first column under “I can”, the cells read “determine the values for which a rational expression is undefined,” “evaluate rational expressions,” “simplify rational expressions,” and “simplify rational expressions with opposite factors.” The rest of the cells are blank.], alt: "This figure shows a table with four columns and five rows. The first row is a header row and each column is labeled. The first column header is labeled “I can…”, the second is labeled “Confidently”, the third is labeled “With some help”, and the fourth is labeled “No—I don’t get it!” In the first column under “I can”, the cells read “determine the values for which a rational expression is undefined,” “evaluate rational expressions,” “simplify rational expressions,” and “simplify rational expressions with opposite factors.” The rest of the cells are blank.", 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.