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CHAPTER 15 REVIEW EXERCISES

CONCEPT CHECK EXERCISES

Determine each of the following as being either true or false. If it is false, explain why.

If $3 x^{2} + 5 x - 8$ $3 x^{2} + 5 x - 8$ is divided by $x - 2,$ $x - 2,$ the remainder is 12.
Using synthetic division to divide $2 x^{3} - 3 x^{2} - 23 x + 12$ $2 x^{3} - 3 x^{2} - 23 x + 12$ by $x + 3,$ $x + 3,$ the bottom row of numbers is 2 $- 9 - 4 0 .$ $- 9 - 4 0 .$
Without solving, it can be determined that 1/8 is a possible rational root of the equation $4 x^{4} - 3 x^{3} + 5 x^{2} - x + 8 = 0 .$ $4 x^{4} - 3 x^{3} + 5 x^{2} - x + 8 = 0 .$
Without solving, it can be determined that there is no more than one possible negative root of the equation $x^{4} - 3 x^{2} + x + 1 = 0 .$ $x^{4} - 3 x^{2} + x + 1 = 0 .$

PRACTICE AND APPLICATIONS

In Exercises 5–8, find the remainder of the indicated division by the remainder theorem.

$(2 x^{3} - 4 x^{2} - x + 7) \div (x - 1)$ $(2 x^{3} - 4 x^{2} - x + 7) \div (x - 1)$
$(x^{3} - 2 x^{2} + 9) \div (x + 2)$ $(x^{3} - 2 x^{2} + 9) \div (x + 2)$
$(3 n^{3} + n + 4) \div (n + 3)$ $(3 n^{3} + n + 4) \div (n + 3)$
$(x^{4} - 5 x^{3} + 8 x^{2} + 15 x - 2) \div (x - 3)$ $(x^{4} - 5 x^{3} + 8 x^{2} + 15 x - 2) \div (x - 3)$

In Exercises 9–12, use the factor theorem to determine whether or not the second expression is a factor of the first.

$x^{4} + x^{3} + x^{2} - 2 x - 3; x + 1$ $x^{4} + x^{3} + x^{2} - 2 x - 3; x + 1$
$2 s^{3} - 6 s - 4; s - 2$ $2 s^{3} - 6 s - 4; s - 2$
$2 t^{4} - 10 t^{3} - t^{2} - 3 t + 10; t + 5$ $2 t^{4} - 10 t^{3} - t^{2} - 3 t + 10; t + 5$
$9 v^{3} + 6 v^{2} + 4 v + 2; 3 v + 1$ $9 v^{3} + 6 v^{2} + 4 v + 2; 3 v + 1$

In Exercises 13–20, use synthetic division to perform the indicated divisions.

$(x^{3} + 4 x^{2} + 5 x + 1) \div (x - 1)$ $(x^{3} + 4 x^{2} + 5 x + 1) \div (x - 1)$
$(3 x^{3} - 2 x^{2} + 7) \div (x - 3)$ $(3 x^{3} - 2 x^{2} + 7) \div (x - 3)$
$(2 x^{3} - 3 x^{2} - 4 x + 3) \div (x + 2)$ $(2 x^{3} - 3 x^{2} - 4 x + 3) \div (x + 2)$
$(3 D^{3} + 8 D^{2} - 16) \div (D + 4)$ $(3 D^{3} + 8 D^{2} - 16) \div (D + 4)$
$(x^{4} + 3 x^{3} - 20 x^{2} - 2 x + 56) \div (x + 6)$ $(x^{4} + 3 x^{3} - 20 x^{2} - 2 x + 56) \div (x + 6)$
$(x^{4} - 6 x^{3} + x - 8) \div (x - 3)$ $(x^{4} - 6 x^{3} + x - 8) \div (x - 3)$
$(2 m^{5} - 48 m^{3} + m^{2} - 9) \div (m - 5)$ $(2 m^{5} - 48 m^{3} + m^{2} - 9) \div (m - 5)$
$(x^{6} + 63 x^{3} + 5 x^{2} - 9 x - 8) \div (x + 4)$ $(x^{6} + 63 x^{3} + 5 x^{2} - 9 x - 8) \div (x + 4)$

In Exercises 21–24, use synthetic division to determine whether or not the given numbers are zeros of the given functions.

$y^{3} + 4 y^{2} - 9; - 3$ $y^{3} + 4 y^{2} - 9; - 3$
$8 y^{4} - 32 y^{3} - y + 4; 4$ $8 y^{4} - 32 y^{3} - y + 4; 4$
$2 x^{4} - x^{3} + 2 x^{2} + x - 1; - 1, \frac{1}{2}$ $2 x^{4} - x^{3} + 2 x^{2} + x - 1; - 1, \frac{1}{2}$
$6 W^{4} + 9 W^{3} - 2 W^{2} + 6 W - 4; - \frac{3}{2}, - \frac{1}{2}$ $6 W^{4} + 9 W^{3} - 2 W^{2} + 6 W - 4; - \frac{3}{2}, - \frac{1}{2}$

In Exercises 25–36, find all the roots of the given equations, using synthetic division and the given roots.

$x^{3} - 4 x^{2} - 7 x + 10 = 0 (r_{1} = 5)$ $x^{3} - 4 x^{2} - 7 x + 10 = 0 (r_{1} = 5)$
$3 B^{3} - 10 B^{2} + B + 14 = 0 (r_{1} = 2)$ $3 B^{3} - 10 B^{2} + B + 14 = 0 (r_{1} = 2)$
$x^{4} - 10 x^{3} + 35 x^{2} - 50 x + 24 = 0 (r_{1} = 1, r_{2} = 2)$ $x^{4} - 10 x^{3} + 35 x^{2} - 50 x + 24 = 0 (r_{1} = 1, r_{2} = 2)$
$2 x^{4} - 2 x^{3} - 10 x^{2} - 2 x - 12 = 0 (r_{1} = 3, r_{2} = - 2)$ $2 x^{4} - 2 x^{3} - 10 x^{2} - 2 x - 12 = 0 (r_{1} = 3, r_{2} = - 2)$
$4 p^{4} - p^{2} - 18 p + 9 = 0 (r_{1} = \frac{1}{2}, r_{2} = \frac{3}{2})$ $4 p^{4} - p^{2} - 18 p + 9 = 0 (r_{1} = \frac{1}{2}, r_{2} = \frac{3}{2})$
$15 x^{4} + 4 x^{3} + 56 x^{2} + 16 x - 16 = 0 (r_{1} = 2/5, r_{2} = - 2/3)$ $15 x^{4} + 4 x^{3} + 56 x^{2} + 16 x - 16 = 0 (r_{1} = 2/5, r_{2} = - 2/3)$
$4 x^{4} + 4 x^{3} + x^{2} + 4 x - 3 = 0 (r_{1} = j)$ $4 x^{4} + 4 x^{3} + x^{2} + 4 x - 3 = 0 (r_{1} = j)$
$x^{4} + 2 x^{3} - 4 x - 4 = 0 (r_{1} = - 1 + j)$ $x^{4} + 2 x^{3} - 4 x - 4 = 0 (r_{1} = - 1 + j)$
$s^{5} + 3 s^{4} - s^{3} - 11 s^{2} - 12 s - 4 = 0$ $s^{5} + 3 s^{4} - s^{3} - 11 s^{2} - 12 s - 4 = 0$ ( $- 1$ $- 1$ is a triple root)
$\begin{array}{l} 24 x^{5} + 10 x^{4} + 7 x^{2} - 6 x + 1 = 0 \\ (r_{1} = - 1, r_{2} = \frac{1}{4}, r_{3} = \frac{1}{3}) \end{array}$ $\begin{array}{l} 24 x^{5} + 10 x^{4} + 7 x^{2} - 6 x + 1 = 0 \\ (r_{1} = - 1, r_{2} = \frac{1}{4}, r_{3} = \frac{1}{3}) \end{array}$
$\begin{array}{l} V^{5} + 4 V^{4} + 5 V^{3} - V^{2} - 4 V - 5 = 0 \\ (r_{1} = 1, r_{2} = - 2 + j) \end{array}$ $\begin{array}{l} V^{5} + 4 V^{4} + 5 V^{3} - V^{2} - 4 V - 5 = 0 \\ (r_{1} = 1, r_{2} = - 2 + j) \end{array}$
$2 x^{6} - x^{5} + 8 x^{2} - 4 x = 0 (r_{1} = \frac{1}{2}, r_{2} = 1 + j)$ $2 x^{6} - x^{5} + 8 x^{2} - 4 x = 0 (r_{1} = \frac{1}{2}, r_{2} = 1 + j)$

In Exercises 37–44, solve the given equations.

$x^{3} + x^{2} - 10 x + 8 = 0$ $x^{3} + x^{2} - 10 x + 8 = 0$
$x^{3} - 8 x^{2} + 20 x = 16$ $x^{3} - 8 x^{2} + 20 x = 16$
$6 r^{3} - 9 r^{2} + 3 = 0$ $6 r^{3} - 9 r^{2} + 3 = 0$
$2 x^{3} - 3 x^{2} - 11 x + 6 = 0$ $2 x^{3} - 3 x^{2} - 11 x + 6 = 0$
$6 x^{3} - x^{2} - 12 x = 5$ $6 x^{3} - x^{2} - 12 x = 5$
$6 y^{3} + 19 y^{2} + 2 y = 3$ $6 y^{3} + 19 y^{2} + 2 y = 3$
$4 t^{4} - 17 t^{2} + 14 t - 3 = 0$ $4 t^{4} - 17 t^{2} + 14 t - 3 = 0$
$2 x^{4} + 5 x^{3} - 14 x^{2} - 23 x + 30 = 0$ $2 x^{4} + 5 x^{3} - 14 x^{2} - 23 x + 30 = 0$

In Exercises 45–70, solve the given problems. Where appropriate, set up the required equations.

Graph the function $f (x) = 3 x^{4} - 7 x^{3} - 26 x^{2} + 16 x + 32,$ $f (x) = 3 x^{4} - 7 x^{3} - 26 x^{2} + 16 x + 32,$ and use the graph as an assist in factoring the function.
Graph the function $f (x) = 8 x^{4} - 22 x^{3} - 11 x^{2} + 52 x - 12,$ $f (x) = 8 x^{4} - 22 x^{3} - 11 x^{2} + 52 x - 12,$ and use the graph as an assist in factoring the function.
What are the possible number of real zeros (double roots count as two, etc.) for a polynomial with real coefficients and of degree 5?
What are the possible combinations of real and nonreal complex zeros (double roots count as two, etc.) of a fourth-degree polynomial?
If a calculator shows a real root, how many nonreal complex roots are possible for a sixth-degree polynomial equation $f (x) = 0 ?$ $f (x) = 0 ?$
Find rational values of a such that $(x - a)$ $(x - a)$ will divide into $x^{3} - 13 x + 3$ $x^{3} - 13 x + 3$ with a remainder of $- 9 .$ $- 9 .$
Explain how to find k if $x + 2$ $x + 2$ is a factor of $f (x) = 3 x^{3} + k x^{2} - 8 x - 8 .$ $f (x) = 3 x^{3} + k x^{2} - 8 x - 8 .$ What is k?
Explain how to find k if $x - 3$ $x - 3$ is a factor of $f (x) = k x^{4} - 15 x^{2} - 5 x - 12 .$ $f (x) = k x^{4} - 15 x^{2} - 5 x - 12 .$ What is k?
Form a polynomial equation of degree 3 with integer coefficients and having roots of j and 5.
If $f (x) = 3 x^{4} - 18 x^{3} - 2 x^{2} + 13 x - 6,$ $f (x) = 3 x^{4} - 18 x^{3} - 2 x^{2} + 13 x - 6,$ and $f (x) = g (x) (x - 6),$ $f (x) = g (x) (x - 6),$ find g(x).
Solve the following system algebraically: $x^{2} = y + 3; x y = 2$ $x^{2} = y + 3; x y = 2$
Where does the graph of the function $f (x) = 2 x^{4} - 7 x^{3} + 11 x^{2} - 28 x + 12$ $f (x) = 2 x^{4} - 7 x^{3} + 11 x^{2} - 28 x + 12$ cross the x-axis?
A silo is to be constructed in the shape of a cylinder with a hemisphere as its top. Because of design constraints, the total height is to be 40.0 ft. Find the radius that would be required in order for the silo to hold ${15,500 ft}^{3}$ ${15,500 ft}^{3}$ of wood chips. Solve graphically.
The edge of a cube is 10 cm greater than the radius of a sphere. If the volumes of the figures are equal, what is this volume?
A computer analysis of the number of crimes committed each month in a certain city for the first 10 months of a year showed that $n = x^{3} - 9 x^{2} + 15 x + 600 .$ $n = x^{3} - 9 x^{2} + 15 x + 600 .$ Here, n is the number of monthly crimes and x is the number of the month (as of the last day). In what month were 580 crimes committed?
A company determined that the number s (in thousands) of computer chips that it could supply at a price p of less than $5 is given by $s = 4 p^{2} - 25,$ $s = 4 p^{2} - 25,$ whereas the demand d (in thousands) for the chips is given by $d = p^{3} - 22 p + 50 .$ $d = p^{3} - 22 p + 50 .$ For what price is the supply equal to the demand?
In order to find the diameter d (in cm) of a helical spring subject to given forces, it is necessary to solve the equation $64 d^{3} - 144 d^{2} + 108 d - 27 = 0 .$ $64 d^{3} - 144 d^{2} + 108 d - 27 = 0 .$ Solve for d.
A cubical tablet for purifying water is wrapped in a sheet of foil 0.500 mm thick. The total volume of the tablet and foil is 33.1% greater than the volume of the tablet alone. Find the length of the edge of the tablet.
For the mirror shown in Fig. 15.13, the reciprocal of the focal distance f equals the sum of the reciprocals of the object distance p and image distance q (in in.). Find p, if $q = p + 4$ $q = p + 4$ and $f = (p + 1) / p .$ $f = (p + 1) / p .$

Fig. 15.13

Figure 15.13 Full Alternative Text
Three electric capacitors are connected in series. The capacitance of the second is $1 μ F$ $1 μ F$ more than the first, and the third is $2 μ F$ $2 μ F$ more than the second. The capacitance of the combination is $1.33 μ F .$ $1.33 μ F .$ The equation used to determine C, the capacitance of the first capacitor, is

$\frac{1}{C} + \frac{1}{C + 1} + \frac{1}{C + 3} = \frac{3}{4}$ $\frac{1}{C} + \frac{1}{C + 1} + \frac{1}{C + 3} = \frac{3}{4}$

Find the values of the capacitances.
The height of a cylindrical oil tank is 3.2 m more than the radius. If the volume of the tank is ${680 m}^{3},$ ${680 m}^{3},$ what are the radius and the height of the tank?
A grain storage bin has a square base, each side of which is 5.5 m longer than the height of the bin. If the bin holds ${160 m}^{3}$ ${160 m}^{3}$ of grain, find its dimensions.
A rectangular door has a diagonal brace that is 0.900 ft longer than the height of the door. If the area of the door is ${24.3 ft}^{2},$ ${24.3 ft}^{2},$ find its dimensions.
The radius of one ball bearing is 1.0 mm greater than the radius of a second ball bearing. If the sum of their volumes is ${100 mm}^{3},$ ${100 mm}^{3},$ find the radius of each.
The entrance to a garden area is a parabolic portal that can be described by $y = 4 - x^{2}$ $y = 4 - x^{2}$ (in m). Find the largest area of a rectangular gate that can be installed by graphing the function for area and finding its maximum. This is similar to finding the maximum point of a parabola as in Section 7.4.
An open container (no top) is to be made from a square piece of sheet metal, 20.0 cm on a side, by cutting equal squares from the corners and bending up the sides. Find the side of each cut-out square such that the volume is a maximum (see Exercise 69.)
A computer science student is to write a computer program that will print out the values of n for which $x + r$ $x + r$ is a factor of $x^{n} + r^{n} .$ $x^{n} + r^{n} .$ Write a paragraph that states which are the values of n and explains how they are found.

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Table of Contents for CHAPTER 15 REVIEW EXERCISES

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Table of Contents for
CHAPTER 15 REVIEW EXERCISES