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== 7.6#h(0.6em)Chapter Summary and Review


    
      
=== Key Concepts


      + The degree of a product of nonzero polynomials is the sum of the degrees of the factors.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Cube of a Binomial]

+ #math.equation(block: false, alt: "open parenthesis x plus y close parenthesis cubed equals x cubed plus 3 x squared y plus 3 x y squared plus y cubed")[$( x + y )^(3) = x^(3) + 3 x^(2) y + 3 x y^(2) + y^(3)$]
+ #math.equation(block: false, alt: "open parenthesis x minus y close parenthesis cubed equals x cubed minus 3 x squared y plus 3 x y squared minus y cubed")[$( x − y )^(3) = x^(3) − 3 x^(2) y + 3 x y^(2) − y^(3)$]
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factoring the Sum or Difference of Two Cubes]

+ #math.equation(block: false, alt: "x cubed plus y cubed equals open parenthesis x plus y close parenthesis open parenthesis x squared minus x y plus y squared close parenthesis")[$x^(3) + y^(3) = ( x + y ) ( x^(2) − x y + y^(2) )$]
+ #math.equation(block: false, alt: "x cubed minus y cubed equals open parenthesis x minus y close parenthesis open parenthesis x squared plus x y plus y squared close parenthesis")[$x^(3) − y^(3) = ( x − y ) ( x^(2) + x y + y^(2) )$]
]
+ The graphs of all polynomials are smooth curves without breaks or holes.
+ The graph of a polynomial of degree #math.equation(block: false, alt: "n")[$n$] (with positive lead coefficient) has the same long-term behavior as the power function of the same degree.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor Theorem]

Let #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] be a polynomial with real number coefficients. Then #math.equation(block: false, alt: "open parenthesis x minus a close parenthesis")[$( x − a )$] is a factor of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] if and only if #math.equation(block: false, alt: "P open parenthesis a close parenthesis equals 0")[$P ( a ) = 0$].
]
+ A polynomial of degree #math.equation(block: false, alt: "n")[$n$] can have at most #math.equation(block: false, alt: "n")[$n$] #math.equation(block: false, alt: "x")[$x$]-intercepts.
+ At a zero of multiplicity #math.equation(block: false, alt: "2")[$2$], the graph of a polynomial has a turning point. At a zero of multiplicity #math.equation(block: false, alt: "3")[$3$], the graph of a polynomial has an inflection point.
+ The square root of a negative number is an imaginary number.
+ A complex number is the sum of a real number and an imaginary number.
+ We can perform the four arithmetic operations on complex numbers
+ The product of a nonzero complex number and its conjugate is always a positive real number.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Fundamental Theorem of Algebra]

Let #math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$] be a polynomial of degree #math.equation(block: false, alt: "n greater than or equal to 1")[$n ≥ 1$]. Then #math.equation(block: false, alt: "p open parenthesis x close parenthesis")[$p ( x )$] has exactly #math.equation(block: false, alt: "n")[$n$] complex zeros.
]
+ We can graph complex numbers in the complex plane
+ Multiplying a complex number by #math.equation(block: false, alt: "i")[$i$] rotates its graph by #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] around the origin.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Rational Function]

A #strong[rational function] is one of the form

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction P open parenthesis x close parenthesis over Q open parenthesis x close parenthesis")[$f ( x ) = frac(P ( x ), Q ( x ))$]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 polynomials and #math.equation(block: false, alt: "Q open parenthesis x close parenthesis")[$Q ( x )$] is not the zero polynomial.
]
+ A rational function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction P open parenthesis x close parenthesis over Q open parenthesis x close parenthesis")[$f ( x ) = display(frac(P ( x ), Q ( x )))$] is undefined for any value #math.equation(block: false, alt: "x equals a")[$x = a$] where #math.equation(block: false, alt: "Q open parenthesis a close parenthesis equals 0")[$Q ( a ) = 0$]. These #math.equation(block: false, alt: "x")[$x$]-values are not in the domain of the function.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Vertical Asymptotes]

If #math.equation(block: false, alt: "Q open parenthesis a close parenthesis equals 0")[$Q ( a ) = 0$] but #math.equation(block: false, alt: "P open parenthesis a close parenthesis not equal to 0")[$P ( a ) ≠ 0$], then the graph of the rational function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction P open parenthesis x close parenthesis over Q open parenthesis x close parenthesis")[$f ( x ) = display(frac(P ( x ), Q ( x )))$] has a #strong[vertical asymptote] at #math.equation(block: false, alt: "x equals a")[$x = a$].
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Horizontal Asymptotes]

Suppose #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction P open parenthesis x close parenthesis over Q open parenthesis x close parenthesis")[$f ( x ) = display(frac(P ( x ), Q ( x )))$] is a rational function, where the degree of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] is #math.equation(block: false, alt: "m")[$m$] and the degree of #math.equation(block: false, alt: "Q open parenthesis x close parenthesis")[$Q ( x )$] is #math.equation(block: false, alt: "n")[$n$].

+ If #math.equation(block: false, alt: "m less than n")[$m < n$], the graph of #math.equation(block: false, alt: "f")[$f$] has a horizontal asymptote at #math.equation(block: false, alt: "y equals 0")[$y = 0$].
+ If #math.equation(block: false, alt: "m equals n")[$m = n$], the graph of #math.equation(block: false, alt: "f")[$f$] has a horizontal asymptote at #math.equation(block: false, alt: "y equals the fraction a over b")[$y = display(frac(a, b))$], where #math.equation(block: false, alt: "a")[$a$] is the lead coefficient of #math.equation(block: false, alt: "P open parenthesis x close parenthesis")[$P ( x )$] and #math.equation(block: false, alt: "b")[$b$] is the lead coefficient of #math.equation(block: false, alt: "Q open parenthesis x close parenthesis")[$Q ( x )$].
+ If #math.equation(block: false, alt: "m greater than n")[$m > n$], the graph of #math.equation(block: false, alt: "f")[$f$] does not have a horizontal asymptote.
]
+ To solve an equation involving an algebraic fraction, we multiply each side of the equation by the denominator of the fraction. This has the effect of clearing the fraction, giving us an equivalent equation without fractions.
+ Whenever we multiply an equation by an expression containing the variable, we should check that the solutions obtained are not extraneous.


    
    
      
=== Chapter 7 Review Problems


      For Problems 1–4, multiply.


      
        #math.equation(block: true, alt: "open parenthesis 2 x minus 5 close parenthesis open parenthesis x squared minus 3 x plus 2 close parenthesis")[$( 2 x − 5 ) ( x^(2) − 3 x + 2 )$]


        
          #math.equation(block: true, alt: "2 x cubed minus 11 x squared plus 19 x minus 10")[$2 x^(3) − 11 x^(2) + 19 x − 10$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis b squared minus 2 b minus 3 close parenthesis open parenthesis 2 b squared plus b minus 5 close parenthesis")[$( b^(2) − 2 b − 3 ) ( 2 b^(2) + b − 5 )$]


      
      
        #math.equation(block: true, alt: "open parenthesis t plus 4 close parenthesis open parenthesis t squared minus t minus 1 close parenthesis")[$( t + 4 ) ( t^(2) − t − 1 )$]


        
          #math.equation(block: true, alt: "t cubed plus 3 t squared minus 5 t minus 4")[$t^(3) + 3 t^(2) − 5 t − 4$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis b plus 3 close parenthesis open parenthesis 2 b minus 1 close parenthesis open parenthesis 2 b plus 5 close parenthesis")[$( b + 3 ) ( 2 b − 1 ) ( 2 b + 5 )$]


      
      For Problems 5–8, find the indicated term.


      
        #math.equation(block: true, alt: "open parenthesis 1 minus 3 x plus 5 x squared close parenthesis open parenthesis 7 plus x minus x squared close parenthesis ; x squared")[$( 1 − 3 x + 5 x^(2) ) ( 7 + x − x^(2) ) ";" " " " " x^(2)$]


        
          #math.equation(block: true, alt: "31 x squared")[$31 x^(2)$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis minus 3 plus x minus 4 x squared close parenthesis open parenthesis 4 plus 3 x minus 2 x cubed close parenthesis ; x cubed")[$( − 3 + x − 4 x^(2) ) ( 4 + 3 x − 2 x^(3) ) ";" " " " " x^(3)$]


      
      
        #math.equation(block: true, alt: "open parenthesis 4 x minus x squared plus 3 x cubed close parenthesis open parenthesis 1 plus 4 x minus 3 x squared close parenthesis ; x cubed")[$( 4 x − x^(2) + 3 x^(3) ) ( 1 + 4 x − 3 x^(2) ) ";" " " " " x^(3)$]


        
          #math.equation(block: true, alt: "minus 13 x cubed")[$− 13 x^(3)$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 3 minus 2 x plus 2 x cubed close parenthesis open parenthesis 5 plus 3 x minus 2 x squared plus 4 x to the power 4 close parenthesis ; x to the power 4")[$( 3 − 2 x + 2 x^(3) ) ( 5 + 3 x − 2 x^(2) + 4 x^(4) ) ";" " " " " x^(4)$]


      
      For Problems 9–12, factor.


      
        #math.equation(block: true, alt: "8 x cubed minus 27 z cubed")[$8 x^(3) − 27 z^(3)$]


        
          #math.equation(block: true, alt: "open parenthesis 2 x minus 3 z close parenthesis open parenthesis 4 x squared plus 6 x z plus 9 z squared close parenthesis")[$( 2 x − 3 z ) ( 4 x^(2) + 6 x z + 9 z^(2) )$]


        
      
      
        #math.equation(block: true, alt: "1 plus 125 a cubed b cubed")[$1 + 125 a^(3) b^(3)$]


      
      
        #math.equation(block: true, alt: "y cubed plus 27 x cubed")[$y^(3) + 27 x^(3)$]


        
          #math.equation(block: true, alt: "open parenthesis y plus 3 x close parenthesis open parenthesis y squared minus 3 x y plus 9 x squared close parenthesis")[$( y + 3 x ) ( y^(2) − 3 x y + 9 x^(2) )$]


        
      
      
        #math.equation(block: true, alt: "x to the power 9 minus 8")[$x^(9) − 8$]


      
      For Problems 13–14, write as a polynomial.


      
        #math.equation(block: true, alt: "open parenthesis v minus 10 close parenthesis cubed")[$( v − 10 )^(3)$]


        
          #math.equation(block: true, alt: "v cubed minus 30 v squared plus 300 v minus 1000")[$v^(3) − 30 v^(2) + 300 v − 1000$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis a plus 2 b squared close parenthesis cubed")[$( a + 2 b^(2) )^(3)$]


      
      
        The expression #math.equation(block: false, alt: "the fraction n over 6 open parenthesis n minus 1 close parenthesis open parenthesis n minus 2 close parenthesis")[$display(frac(n, 6)) ( n − 1 ) ( n − 2 )$] gives the number of different #math.equation(block: false, alt: "3")[$3$]-item pizzas that can be created from a list of #math.equation(block: false, alt: "n")[$n$] toppings.


        + Write the expression as a polynomial.
+ If Mitch's Pizza offers #math.equation(block: false, alt: "12")[$12$] different toppings, how many different combinations for #math.equation(block: false, alt: "3")[$3$]-item pizzas can be made?
+ Use a table or graph to determine how many different toppings are needed in order to be able to have more than #math.equation(block: false, alt: "1000")[$1000$] possible combinations for #math.equation(block: false, alt: "3")[$3$]-item pizzas.


        
          + #math.equation(block: false, alt: "the fraction 1 over 6 n cubed minus the fraction 1 over 2 n squared plus the fraction 1 over 3 n")[$display(frac(1, 6)) n^(3) − display(frac(1, 2)) n^(2) + display(frac(1, 3)) n$]
+ #math.equation(block: false, alt: "220")[$220$]
+ #math.equation(block: false, alt: "20")[$20$]


        
      
      
        The expression #math.equation(block: false, alt: "n open parenthesis n minus 1 close parenthesis open parenthesis n minus 2 close parenthesis")[$n ( n − 1 ) ( n − 2 )$] gives the number of different triple-scoop ice cream cones that can be created from a list of #math.equation(block: false, alt: "n")[$n$] flavors.


        + Write the expression as a polynomial.
+ If Zanner's Ice Cream Parlor offers #math.equation(block: false, alt: "21")[$21$] flavors, how many different triple-scoop ice cream cones can be made?
+ Use a table or graph to determine how many different flavors are needed in order to be able to have more than #math.equation(block: false, alt: "10 , 000")[$10 , 000$] possible triple-scoop ice cream cones.


      
      For Problems 17–18,


      + Graph each polynomial in the standard window.
+ Find the range of the function on the domain #math.equation(block: false, alt: "open bracket minus 10 , 10 close bracket")[$[ − 10 , 10 ]$].


      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x cubed minus 3 x plus 2")[$f ( x ) = x^(3) − 3 x + 2$]


        
          + #figure(figph[cubic], alt: "cubic", caption: none)
+ #math.equation(block: false, alt: "open bracket minus 968 , 972 close bracket")[$[ − 968 , 972 ]$]


        
      
      
        #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals minus 0.1 open parenthesis x to the power 4 minus 6 x cubed plus x squared plus 24 x plus 16 close parenthesis")[$g ( x ) = − 0.1 ( x^(4) − 6 x^(3) + x^(2) + 24 x + 16 )$]


      
      For Problems 19–28,


      + Find the zeros of the polynomial.
+ Sketch the graph by hand.


      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals open parenthesis x minus 2 close parenthesis open parenthesis x plus 1 close parenthesis squared")[$f ( x ) = ( x − 2 ) ( x + 1 )^(2)$]


        
          + #math.equation(block: false, alt: "2 , minus 1")[$2 , − 1$]
+ #figure(figph[cubic], alt: "cubic", caption: none)


        
      
      
        #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals open parenthesis x minus 3 close parenthesis squared open parenthesis x plus 2 close parenthesis")[$g ( x ) = ( x − 3 )^(2) ( x + 2 )$]


      
      
        #math.equation(block: true, alt: "G open parenthesis x close parenthesis equals x squared open parenthesis x minus 1 close parenthesis open parenthesis x plus 3 close parenthesis")[$G ( x ) = x^(2) ( x − 1 ) ( x + 3 )$]


        
          + #math.equation(block: false, alt: "0 , 1 , minus 3")[$0 , 1 , − 3$]
+ #figure(figph[quartic], alt: "quartic", caption: none)


        
      
      
        #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals open parenthesis x plus 1 close parenthesis squared open parenthesis x minus 2 close parenthesis squared")[$F ( x ) = ( x + 1 )^(2) ( x − 2 )^(2)$]


      
      
        #math.equation(block: true, alt: "V open parenthesis x close parenthesis equals x cubed minus x to the power 5")[$V ( x ) = x^(3) − x^(5)$]


        
          + #math.equation(block: false, alt: "0 , 1 , minus 1")[$0 , 1 , − 1$]
+ #figure(figph[quintic], alt: "quintic", caption: none)


        
      
      
        #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals x to the power 4 minus 9 x squared")[$H ( x ) = x^(4) − 9 x^(2)$]


      
      
        #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x cubed plus x squared minus x minus 1")[$P ( x ) = x^(3) + x^(2) − x − 1$]


        
          + #math.equation(block: false, alt: "minus 1 , 1")[$− 1 , 1$]
+ #figure(figph[cubic], alt: "cubic", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals x cubed plus x squared minus 2 x")[$y = x^(3) + x^(2) − 2 x$]


      
      
        #math.equation(block: true, alt: "y equals x to the power 4 minus 7 x squared plus 6")[$y = x^(4) − 7 x^(2) + 6$]


        
          + #math.equation(block: false, alt: "minus 1 , 1 , plus or minus the square root of 6")[$− 1 , 1 , ± sqrt(6)$]
+ #figure(figph[quartic], alt: "quartic", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals x to the power 4 plus x cubed minus 3 x squared minus 3 x")[$y = x^(4) + x^(3) − 3 x^(2) − 3 x$]


      
      For Problems 29–34, find a possible formula for the polynomial, in factored form.


      
        #figure(figph[cubic], alt: "cubic", caption: none)


        
          #math.equation(block: true, alt: "x open parenthesis x plus 2 close parenthesis open parenthesis x minus 3 close parenthesis")[$x ( x + 2 ) ( x − 3 )$]


        
      
      
        #figure(figph[cubic], alt: "cubic", caption: none)


      
      
        #figure(figph[quintic], alt: "quintic", caption: none)


        
          #math.equation(block: true, alt: "x cubed open parenthesis x plus 2 close parenthesis open parenthesis x minus 2 close parenthesis")[$x^(3) ( x + 2 ) ( x − 2 )$]


        
      
      
        #figure(figph[quintic], alt: "quintic", caption: none)


      
      
        #figure(figph[quartic], alt: "quartic", caption: none)


        
          #math.equation(block: true, alt: "x squared open parenthesis x plus 4 close parenthesis open parenthesis x minus 4 close parenthesis")[$x^(2) ( x + 4 ) ( x − 4 )$]


        
      
      
        #figure(figph[quartic], alt: "quartic", caption: none)


      
      For Problems 35–36,


      + Verify that the given value is a zero of the polynomial.
+ Find the other zeros. (#emph[Hint]: Use polynomial division to write #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus a close parenthesis Q open parenthesis x close parenthesis")[$P ( x ) = ( x − a ) Q ( x )$], then factor #math.equation(block: false, alt: "Q open parenthesis x close parenthesis")[$Q ( x )$].)


      
        #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals x cubed minus x squared minus 7 x minus 2")[$P ( x ) = x^(3) − x^(2) − 7 x − 2$]; #math.equation(block: true, alt: "a equals minus 2")[$a = − 2$]


        
          + #math.equation(block: false, alt: "P open parenthesis minus 2 close parenthesis equals 0")[$P ( − 2 ) = 0$]
+ #math.equation(block: false, alt: "the fraction 3 plus or minus the square root of 13 over 2")[$display(frac(3 ± sqrt(13), 2))$]


        
      
      
        #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals 3 x cubed minus 11 x squared minus 5 x plus 4")[$P ( x ) = 3 x^(3) − 11 x^(2) − 5 x + 4$]; #math.equation(block: true, alt: "a equals 4")[$a = 4$]


      
      For Problems 37–40,


      + Solve the quadratic equation, and write the solutions in the form #math.equation(block: false, alt: "a plus b i")[$a + b i$].
+ Check your solutions.


      
        #math.equation(block: true, alt: "x squared plus 4 x plus 10 equals 0")[$x^(2) + 4 x + 10 = 0$]


        
          #math.equation(block: true, alt: "minus 2 plus or minus i the square root of 6")[$− 2 ± i sqrt(6)$]


        
      
      
        #math.equation(block: true, alt: "x squared minus 2 x plus 7 equals 0")[$x^(2) − 2 x + 7 = 0$]


      
      
        #math.equation(block: true, alt: "3 x squared minus 6 x plus 5 equals 0")[$3 x^(2) − 6 x + 5 = 0$]


        
          #math.equation(block: true, alt: "1 plus or minus the fraction the square root of 6 over 3 i")[$1 ± display(frac(sqrt(6), 3)) i$]


        
      
      
        #math.equation(block: true, alt: "2 x squared plus 5 x plus 4 equals 0")[$2 x^(2) + 5 x + 4 = 0$]


      
      For Problems 41–42, evaluate the polynomial for the given values of the variable.


      
        #math.equation(block: true, alt: "z squared minus 6 z plus 5")[$z^(2) − 6 z + 5$]


        + #math.equation(block: false, alt: "z equals 3 plus 2 i")[$z = 3 + 2 i$]
+ #math.equation(block: false, alt: "z equals 3 minus 2 i")[$z = 3 − 2 i$]


        
          + #math.equation(block: false, alt: "minus 8")[$− 8$]
+ #math.equation(block: false, alt: "minus 8")[$− 8$]


        
      
      
        #math.equation(block: true, alt: "w squared plus 4 w plus 7")[$w^(2) + 4 w + 7$]


        + #math.equation(block: false, alt: "w equals minus 1 minus 3 i")[$w = − 1 − 3 i$]
+ #math.equation(block: false, alt: "w equals minus 1 plus 3 i")[$w = − 1 + 3 i$]


      
      For Problems 43–44, find the quotient.


      
        #math.equation(block: true, alt: "the fraction 2 minus 5 i over 3 minus i")[$display(frac(2 − 5 i, 3 − i))$]


        
          #math.equation(block: true, alt: "the fraction 11 over 10 minus the fraction 13 over 10 i")[$display(frac(11, 10)) − display(frac(13, 10)) i$]


        
      
      
        #math.equation(block: true, alt: "the fraction 1 plus i over 1 minus i")[$display(frac(1 + i, 1 − i))$]


      
      For Problems 45–46, find a fourth-degree polynomial with the given zeros.


      
        #math.equation(block: true, alt: "3 i , 1 minus 2 i")[$3 i , " " 1 − 2 i$]


        
          #math.equation(block: true, alt: "x to the power 4 minus 2 x cubed plus 14 x squared minus 18 x plus 45")[$x^(4) − 2 x^(3) + 14 x^(2) − 18 x + 45$]


        
      
      
        #math.equation(block: true, alt: "2 minus the square root of 3 , 2 plus 3 i")[$2 − sqrt(3) , " " 2 + 3 i$]


      
      For Problems 45–46, plot each complex number as a point on the complex plane.


      
        + #math.equation(block: false, alt: "z equals 4 plus i , z bar , minus z , minus z bar")[$z = 4 + i , " " limits(z)^(―) , " " − z , " " − limits(z)^(―)$]
+ #math.equation(block: false, alt: "i z , i z bar , minus i z , minus i z bar")[$i z , " " i limits(z)^(―) , " " − i z , " " − i limits(z)^(―)$]


        
          + #figure(figph[four points in complex plane], alt: "four points in complex plane", caption: none)
+ #figure(figph[four points in complex plane], alt: "four points in complex plane", caption: none)


        
      
      
        + #math.equation(block: false, alt: "w equals minus 2 plus 3 i , w bar , minus w , minus w bar")[$w = − 2 + 3 i , " " limits(w)^(―) , " " − w , " " − limits(w)^(―)$]
+ #math.equation(block: false, alt: "i w , i w bar , minus i w , minus i w bar")[$i w , " " i limits(w)^(―) , " " − i w , " " − i limits(w)^(―)$]


      
      
        The radius, #math.equation(block: false, alt: "r")[$r$], of a cylindrical can should be one-half its height, #math.equation(block: false, alt: "h")[$h$].


        + Express the volume, #math.equation(block: false, alt: "V")[$V$], of the can as a function of its height.
+ What is the volume of the can if its height is #math.equation(block: false, alt: "2")[$2$] centimeters? #math.equation(block: false, alt: "4")[$4$] centimeters?
+ Graph the volume as a function of the height and verify your results of part (b) graphically. What is the approximate height of the can if its volume is #math.equation(block: false, alt: "100")[$100$] cubic centimeters?


        
          + #math.equation(block: false, alt: "V equals the fraction π h cubed over 4")[$V = display(frac(π h^(3), 4))$]
+ #math.equation(block: false, alt: "2 π cm cubed approximately equals 6.28 cm cubed")[$2 π attach(" cm", t: 3) ≈ 6.28 attach(" cm", t: 3)$]; #math.equation(block: false, alt: "16 π cm cubed approximately equals 50.27 cm cubed")[$16 π attach(" cm", t: 3) ≈ 50.27 attach(" cm", t: 3)$]
+ #figure(figph[cubic], alt: "cubic", caption: none)


        
      
      
        The Twisty-Freez machine dispenses soft ice cream in a cone-shaped peak with a height #math.equation(block: false, alt: "3")[$3$] times the radius of its base. The ice cream comes in a round bowl with base diameter #math.equation(block: false, alt: "d")[$d$].


        + Express the volume, #math.equation(block: false, alt: "V")[$V$], of  Twisty-Freez in the bowl as a function of #math.equation(block: false, alt: "d")[$d$].
+ How much Twisty-Freez comes in a #math.equation(block: false, alt: "3")[$3$]-inch diameter dish? A #math.equation(block: false, alt: "4")[$4$]-inch dish?
+ Graph the volume as a function of the diameter and verify your results of part (b) graphically. What is the approximate diameter of a Twisty-Freez if its volume is #math.equation(block: false, alt: "5")[$5$] cubic inches?


      
      
        A new health club opened up, and the manager kept track of the number of active members over its first few months of operation. The equation below gives the number, #math.equation(block: false, alt: "N")[$N$], of active members, in hundreds, #math.equation(block: false, alt: "t")[$t$] months after the club opened.


        #math.equation(block: true, alt: "N equals the fraction 44 t over 40 plus t squared")[$N = frac(44 t, 40 + t^(2))$]
        + Use your calculator to graph the function #math.equation(block: false, alt: "N")[$N$] on a suitable domain.
+ How many active members did the club have after #math.equation(block: false, alt: "8")[$8$] months?
+ In which months did the club have #math.equation(block: false, alt: "200")[$200$] active members?
+ When does the health club have the largest number of active members? What happens to the number of active members as time goes on?


        
          + #figure(figph[rational function], alt: "rational function", caption: none)
+ #math.equation(block: false, alt: "338")[$338$]
+ Months #math.equation(block: false, alt: "2")[$2$] and #math.equation(block: false, alt: "20")[$20$]
+ During month #math.equation(block: false, alt: "6")[$6$]. The number of members eventually decreases to zero.


        
      
      
        A small lake in a state park has become polluted by runoff from a factory upstream. The cost for removing #math.equation(block: false, alt: "p")[$p$] percent of the pollution from the lake is given, in thousands of dollars, by


        #math.equation(block: true, alt: "C equals the fraction 25 p over 100 minus p")[$C = frac(25 p, 100 − p)$]
        + Use your calculator to graph the function #math.equation(block: false, alt: "C")[$C$] on a suitable domain.
+ How much will it cost to remove #math.equation(block: false, alt: "40 %")[$40 upright(%)$] of the pollution?
+ How much of the pollution can be removed for #math.equation(block: false, alt: "$ 100 , 000")[$upright(\$) 100 , 000$]?
+ What happens to the cost as the amount of pollution to be removed increases? How much will it cost to remove all the pollution?


      
      For Problems 53–54, state the domain of the function.


      
        #math.equation(block: true, alt: "h open parenthesis x close parenthesis equals the fraction x squared minus 9 over x open parenthesis x squared minus 4 close parenthesis")[$h ( x ) = display(frac(x^(2) − 9, x ( x^(2) − 4 )))$]


        
          All numbers except #math.equation(block: false, alt: "minus 2 , 0 , 2")[$− 2 , 0 , 2$].


        
      
      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction x squared minus 3 x plus 10 over x squared open parenthesis x squared plus 1 close parenthesis")[$f ( x ) = display(frac(x^(2) − 3 x + 10, x^(2) ( x^(2) + 1 )))$]


      
      For Problems 55–56,


      + Sketch the horizontal and vertical asymptotes for each function.
+ Use the asymptotes to help you sketch the graph.


      
        #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals the fraction 2 x over x squared minus 1")[$F ( x ) = display(frac(2 x, x^(2) − 1))$]


        
          #figure(figph[rational function], alt: "rational function", caption: none)


        
      
      
        #math.equation(block: true, alt: "G open parenthesis x close parenthesis equals the fraction 2 over x squared minus 1")[$G ( x ) = display(frac(2, x^(2) − 1))$]


      
      For Problems 57–62,


      + Identify all asymptotes and intercepts.
+ Sketch the graph.


      
        #math.equation(block: true, alt: "y equals the fraction 1 over x minus 4")[$y = display(frac(1, x − 4))$]


        
          + Horizontal asymptote #math.equation(block: false, alt: "y equals 0")[$y = 0$]; Vertical asymptote #math.equation(block: false, alt: "x equals 4")[$x = 4$]; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , the fraction minus 1 over 4 close parenthesis")[$( 0 , frac(− 1, 4) )$]
+ #figure(figph[translation of reciprocal], alt: "translation of reciprocal", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals the fraction 2 over x squared minus 3 x minus 10")[$y = display(frac(2, x^(2) − 3 x − 10))$]


      
      
        #math.equation(block: true, alt: "y equals the fraction x minus 2 over x plus 3")[$y = display(frac(x − 2, x + 3))$]


        
          + Horizontal asymptote #math.equation(block: false, alt: "y equals 1")[$y = 1$]; Vertical asymptote #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$]; #math.equation(block: false, alt: "x")[$x$]-intercept #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , the fraction minus 2 over 3 close parenthesis")[$( 0 , frac(− 2, 3) )$]
+ #figure(figph[translation of reciprocal], alt: "translation of reciprocal", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals the fraction x minus 1 over x squared minus 2 x minus 3")[$y = display(frac(x − 1, x^(2) − 2 x − 3))$]


      
      
        #math.equation(block: true, alt: "y equals the fraction 3 x squared over x squared minus 4")[$y = display(frac(3 x^(2), x^(2) − 4))$]


        
          + Horizontal asymptote #math.equation(block: false, alt: "y equals 3")[$y = 3$]; Vertical asymptote #math.equation(block: false, alt: "x equals plus or minus 2")[$x = ± 2$]; #math.equation(block: false, alt: "x")[$x$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]
+ #figure(figph[translation of reciprocal], alt: "translation of reciprocal", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals the fraction 2 x squared minus 2 over x squared minus 9")[$y = display(frac(2 x^(2) − 2, x^(2) − 9))$]


      
      For Problems 63–66,


      + Use polynomial division to write the fraction in the form #math.equation(block: false, alt: "y equals the fraction k over p open parenthesis x close parenthesis plus c")[$y = display(frac(k, p ( x ))) + c$], where #math.equation(block: false, alt: "k")[$k$] and #math.equation(block: false, alt: "c")[$c$] are constants.
+ Use transformations to sketch the graph.


      
        #math.equation(block: true, alt: "y equals the fraction 3 x plus 4 over x plus 3")[$y = display(frac(3 x + 4, x + 3))$]


        
          + #math.equation(block: false, alt: "y equals the fraction minus 5 over x plus 3 plus 3")[$y = display(frac(− 5, x + 3)) + 3$]
+ #figure(figph[transformation of reciprocal], alt: "transformation of reciprocal", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals the fraction 5 x plus 1 over x minus 2")[$y = display(frac(5 x + 1, x − 2))$]


      
      
        #math.equation(block: true, alt: "y equals the fraction x squared plus 2 x plus 3 over open parenthesis x plus 1 close parenthesis squared")[$y = display(frac(x^(2) + 2 x + 3, ( x + 1 )^(2)))$]


        
          + #math.equation(block: false, alt: "y equals the fraction 2 over open parenthesis x plus 1 close parenthesis squared plus 1")[$y = display(frac(2, ( x + 1 )^(2))) + 1$]
+ #figure(figph[transformation of reciprocal-squared], alt: "transformation of reciprocal-squared", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals the fraction x squared minus 4 x plus 3 over open parenthesis x minus 2 close parenthesis squared")[$y = display(frac(x^(2) − 4 x + 3, ( x − 2 )^(2)))$]


      
      
        The Explorer's Club is planning a canoe trip to travel #math.equation(block: false, alt: "90")[$90$] miles up the Lazy River and return in #math.equation(block: false, alt: "4")[$4$] days. Club members plan to paddle for #math.equation(block: false, alt: "6")[$6$] hours each day, and they know that the current in the Lazy River is #math.equation(block: false, alt: "2")[$2$] miles per hour.


        + Express the time it will take for the upstream journey as a function of their paddling speed in still water.
+ Express the time it will take for the downstream journey as a function of their paddling speed in still water.
+ Graph the sum of the two functions and find the point on the graph with #math.equation(block: false, alt: "y")[$y$]-coordinate #math.equation(block: false, alt: "24")[$24$]. Interpret the coordinates of the point in the context of the problem.
+ The Explorer's Club would like to know what average paddling speed members must maintain in order to complete their trip in #math.equation(block: false, alt: "4")[$4$] days. Write an equation to describe this situation.
+ Solve your equation to find the required paddling speed.


        
          + #math.equation(block: false, alt: "t sub 1 equals the fraction 90 over v minus 2")[$t_(1) = display(frac(90, v − 2))$]
+ #math.equation(block: false, alt: "t sub 2 equals the fraction 90 over v plus 2")[$t_(2) = display(frac(90, v + 2))$]
+ #figure(figph[curve], alt: "curve", caption: none)
+ #math.equation(block: false, alt: "the fraction 90 over v minus 2 plus the fraction 90 over v plus 2 equals 24")[$display(frac(90, v − 2)) + display(frac(90, v + 2)) = 24$]
+ #math.equation(block: false, alt: "8")[$8$] mph


        
      
      
        Pam lives on the banks of the Cedar River and makes frequent trips in her outboard motorboat. The boat travels at #math.equation(block: false, alt: "20")[$20$] miles per hour in still water.


        + Express the time it takes Pam to travel #math.equation(block: false, alt: "8")[$8$] miles upstream to the gas station as a function of the speed of the current.
+ Express the time it takes Pam to travel #math.equation(block: false, alt: "12")[$12$] miles downstream to Marie's house as a function of the speed of the current.
+ Graph the two functions in the same window, then find the coordinates of the intersection point. Interpret those coordinates in the context of the problem.
+ Pam traveled to the gas station in the same time it took her to travel to Marie's house. Write an equation to describe this situation.
+ Solve your equation to find the speed of the current in the Cedar River.


      
      
        Mikala sells #math.equation(block: false, alt: "the fraction 320 over x")[$display(frac(320, x))$] bottles of bath oil per week if she charges #math.equation(block: false, alt: "x")[$x$] dollars per bottle. Her supplier can manufacture #math.equation(block: false, alt: "the fraction 1 over 2 x plus 6")[$display(frac(1, 2)) x + 6$] bottles per week if she sells it at #math.equation(block: false, alt: "x")[$x$] dollars per bottle.


        + Graph the demand function, #math.equation(block: false, alt: "D open parenthesis x close parenthesis equals the fraction 320 over x")[$D ( x ) = display(frac(320, x))$], and the supply function, #math.equation(block: false, alt: "S open parenthesis x close parenthesis equals the fraction 1 over 2 x plus 6")[$S ( x ) = display(frac(1, 2)) x + 6$], in the same window.
+ Write and solve an equation to find the equilibrium price, that is, the price at which the supply equals the demand for bath oil. Label this point on your graph.


        
          + #figure(figph[supply and demand curves], alt: "supply and demand curves", caption: none)
+ #math.equation(block: false, alt: "the fraction 320 over x equals the fraction 1 over 2 x plus 6")[$display(frac(320, x)) = display(frac(1, 2)) x + 6 " "$]; \$#math.equation(block: false, alt: "20")[$20$]


        
      
      
        Tomoko sells #math.equation(block: false, alt: "the fraction 4800 over x")[$display(frac(4800, x))$] exercise machines each month if the price of a machine is #math.equation(block: false, alt: "x")[$x$] dollars. On the other hand, her supplier can manufacture #math.equation(block: false, alt: "2.5 x plus 20")[$2.5 x + 20$] machines if she charges #math.equation(block: false, alt: "x")[$x$] dollars apiece for them.


        + Graph the demand function, #math.equation(block: false, alt: "D open parenthesis x close parenthesis equals the fraction 4800 over x")[$D ( x ) = display(frac(4800, x))$], and the supply function, #math.equation(block: false, alt: "S open parenthesis x close parenthesis equals 2.5 x plus 20")[$S ( x ) = 2.5 x + 20$], in the same window.
+ Write and solve an equation to find the equilibrium price, that is, the price at which the supply equals the demand for exercise machines. Label this point on your graph.


      
      For Problems 71–72, write and solve a proportion.


      
        A polling firm finds that #math.equation(block: false, alt: "78")[$78$] of the #math.equation(block: false, alt: "300")[$300$] randomly selected students at Citrus College play some musical instrument. Based on the poll, how many of the college’s #math.equation(block: false, alt: "1150")[$1150$] students play a musical instrument?


        
          #math.equation(block: true, alt: "299")[$299$]


        
      
      
        Claire wants to make a scale model of Salem College. The largest building on campus, Lausanne Hall, is #math.equation(block: false, alt: "60")[$60$] feet tall, and her model of Lausanne Hall will be #math.equation(block: false, alt: "8")[$8$] inches tall. How tall should she make the model of Willamette Hall, which is #math.equation(block: false, alt: "48")[$48$] feet tall?


      
      For Problems 73–80, solve.


      
        #math.equation(block: true, alt: "the fraction y plus 3 over y plus 5 equals the fraction 1 over 3")[$display(frac(y + 3, y + 5)) = display(frac(1, 3))$]


        
          #math.equation(block: true, alt: "minus 2")[$− 2$]


        
      
      
        #math.equation(block: true, alt: "the fraction z squared plus 2 over z squared minus 2 equals 3")[$display(frac(z^(2) + 2, z^(2) − 2)) = 3$]


      
      
        #math.equation(block: true, alt: "the fraction x over x minus 2 equals the fraction 2 over x minus 2 plus 7")[$display(frac(x, x − 2)) = display(frac(2, x − 2)) + 7$]


        
          No solution


        
      
      
        #math.equation(block: true, alt: "the fraction 3 x over x plus 1 minus the fraction 2 over x squared plus x equals the fraction 4 over x")[$display(frac(3 x, x + 1)) − display(frac(2, x^(2) + x)) = display(frac(4, x))$]


      
      
        #math.equation(block: true, alt: "the fraction 2 over a plus 1 plus the fraction 1 over a minus 1 equals the fraction 3 a minus 1 over a squared minus 1")[$display(frac(2, a + 1)) + display(frac(1, a − 1)) = display(frac(3 a − 1, a^(2) − 1))$]


        
          All #math.equation(block: false, alt: "a")[$a$] except #math.equation(block: false, alt: "minus 1")[$− 1$] and #math.equation(block: false, alt: "1")[$1$]


        
      
      
        #math.equation(block: true, alt: "the fraction 2 b minus 1 over b squared plus 2 b equals the fraction 4 over b plus 2 minus the fraction 1 over b")[$display(frac(2 b − 1, b^(2) + 2 b)) = display(frac(4, b + 2)) − display(frac(1, b))$]


      
      
        #math.equation(block: true, alt: "the fraction minus 10 over u minus 2 equals the fraction u minus 4 over u squared minus u minus 2 plus the fraction 3 over u plus 1")[$display(frac(− 10, u − 2)) = display(frac(u − 4, u^(2) − u − 2)) + display(frac(3, u + 1))$]


        
          #math.equation(block: true, alt: "0")[$0$]


        
      
      
        #math.equation(block: true, alt: "the fraction 1 over t squared plus t plus the fraction 1 over t equals the fraction 3 over t plus 1")[$display(frac(1, t^(2) + t)) + display(frac(1, t)) = display(frac(3, t + 1))$]


      
      For Problems 81–84, solve for the indicated variable.


      
        #math.equation(block: false, alt: "V equals C open parenthesis 1 minus the fraction t over n close parenthesis")[$V = C ( 1 − display(frac(t, n)) )$], for #math.equation(block: false, alt: "n")[$n$]


        
          #math.equation(block: true, alt: "n equals the fraction C t over C minus V")[$n = display(frac(C t, C − V))$]


        
      
      
        #math.equation(block: false, alt: "r equals the fraction d c over 1 minus e c")[$r = display(frac(d c, 1 − e c))$], for #math.equation(block: false, alt: "c")[$c$]


      
      
        #math.equation(block: false, alt: "the fraction p over q equals the fraction r over q plus r")[$display(frac(p, q)) = display(frac(r, q + r))$], for #math.equation(block: false, alt: "q")[$q$]


        
          #math.equation(block: true, alt: "q equals the fraction p r over r minus p")[$q = display(frac(p r, r − p))$]


        
      
      
        #math.equation(block: false, alt: "I equals the fraction E over R plus the fraction r over n")[$I = display(frac(E, R + display(frac(r, n))))$], for #math.equation(block: false, alt: "R")[$R$]