Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 15

Equations of Higher Degree - all with Video Answers

Educators

Section 1

The Remainder and Factor Theorems; Synthetic Division

Problem 1

Make the given changes in the indicated examples of this section, and then perform the indicated operations.
In Example $3,$ change the $x+4$ to $x+3$ and then find the remainder.

Dwijendra Rao

Dwijendra Rao

Numerade Educator

Problem 2

Make the given changes in the indicated examples of this section, and then perform the indicated operations.
In Example $4(a),$ change the $t+1$ to $t-1$ and then determine if $t-1$ is a factor.

Dwijendra Rao

Numerade Educator

Problem 3

Make the given changes in the indicated examples of this section, and then perform the indicated operations.
In Example $6,$ change the $x+3$ to $x+2$ and then perform the synthetic division.

Dwijendra Rao

Numerade Educator

Problem 4

Make the given changes in the indicated examples of this section, and then perform the indicated operations.
In Example $9,$ change the $2 x-3$ to $2 x+3$ and then determine whether $2 x+3$ is a factor.

Dwijendra Rao

Numerade Educator

Problem 5

Find the remainder by long division.
$$\left(x^{3}+2 x-8\right) \div(x-2)$$

Dwijendra Rao

Numerade Educator

Problem 6

Find the remainder by long division.
$$\left(x^{4}-4 x^{3}-x^{2}+x-100\right) \div(x+3)$$

Dwijendra Rao

Numerade Educator

Problem 7

Find the remainder by long division.
$$\left(2 x^{5}-x^{2}+8 x+44\right) \div(x+1)$$

Dwijendra Rao

Numerade Educator

Problem 8

Find the remainder by long division.
$$\left(4 s^{3}-9 s^{2}-24 s-17\right) \div(s-5)$$

Dwijendra Rao

Numerade Educator

Problem 9

Find the remainder by long division.
$$\left(2 x^{4}-3 x^{3}-2 x^{2}-15 x-16\right) \div(2 x-3)$$

Dwijendra Rao

Numerade Educator

Problem 10

Find the remainder by long division.
$$\left(2 x^{4}-11 x^{2}-15 x-17\right) \div(2 x+1)$$

Dwijendra Rao

Numerade Educator

Problem 11

Find the remainder using the remainder theorem. Do not use synthetic division.
$$\left(R^{4}+R^{3}-9 R^{2}+3\right) \div(R-3)$$

Dwijendra Rao

Numerade Educator

Problem 12

Find the remainder using the remainder theorem. Do not use synthetic division.
$$\left(4 x^{4}-x^{3}+5 x-7\right) \div(x-5)$$

Dwijendra Rao

Numerade Educator

Problem 13

Find the remainder using the remainder theorem. Do not use synthetic division.
$$\left(2 x^{4}-7 x^{3}-x^{2}+8\right) \div(x+1)$$

Dwijendra Rao

Numerade Educator

Problem 14

Find the remainder using the remainder theorem. Do not use synthetic division.
$$\left(3 n^{4}-13 n^{2}+10 n-10\right) \div(n+4)$$

Dwijendra Rao

Numerade Educator

Problem 15

Find the remainder using the remainder theorem. Do not use synthetic division.
$$\left(x^{5}-3 x^{3}+5 x^{2}-10 x+6\right) \div(x+2)$$

Dwijendra Rao

Numerade Educator

Problem 16

Find the remainder using the remainder theorem. Do not use synthetic division.
$$\left(3 x^{4}-12 x^{3}-60 x+4\right) \div(x-0.5)$$

Dwijendra Rao

Numerade Educator

Problem 17

Use the factor theorem to determine whether or not the second expression is a factor of the first expression. Do not use synthetic division.
$$8 x^{3}+2 x^{2}-32 x-8, x-2$$

Dwijendra Rao

Numerade Educator

Problem 18

Use the factor theorem to determine whether or not the second expression is a factor of the first expression. Do not use synthetic division.
$$3 x^{3}+14 x^{2}+7 x-4, x+4$$

Dwijendra Rao

Numerade Educator

Problem 19

Use the factor theorem to determine whether or not the second expression is a factor of the first expression. Do not use synthetic division.
$$3 V^{4}-7 V^{3}+V+8, V-2$$

Dwijendra Rao

Numerade Educator

Problem 20

Use the factor theorem to determine whether or not the second expression is a factor of the first expression. Do not use synthetic division.
$$x^{5}-2 x^{4}+3 x^{3}-6 x^{2}-4 x+8, x-1$$

Dwijendra Rao

Numerade Educator

Problem 21

Use the factor theorem to determine whether or not the second expression is a factor of the first expression. Do not use synthetic division.
$$x^{51}-2 x-1, x+1$$

Dwijendra Rao

Numerade Educator

Problem 22

Use the factor theorem to determine whether or not the second expression is a factor of the first expression. Do not use synthetic division.
$$x^{7}-128^{-1}, x+2^{-1}$$

Dwijendra Rao

Numerade Educator

Problem 23

Perform the indicated divisions by synthetic division.
$$2 x^{5}-x^{3}+3 x^{2}-4 ; \quad x+1$$

Dwijendra Rao

Numerade Educator

Problem 24

Perform the indicated divisions by synthetic division.
$$\left(x^{3}-3 x^{2}-x+2\right) \div(x-3)$$

Dwijendra Rao

Numerade Educator

Problem 25

Perform the indicated divisions by synthetic division.
$$\left(x^{3}+2 x^{2}-3 x+4\right) \div(x+4)$$

Dwijendra Rao

Numerade Educator

Problem 26

Perform the indicated divisions by synthetic division.
$$\left(2 x^{3}-4 x^{2}+x-1\right) \div(x+2)$$

Dwijendra Rao

Numerade Educator

Problem 27

Perform the indicated divisions by synthetic division.
$$\left(p^{6}-6 p^{3}-2 p^{2}-6\right) \div(p-2)$$

Dwijendra Rao

Numerade Educator

Problem 28

Perform the indicated divisions by synthetic division.
$$\left(x^{5}+4 x^{4}-8\right) \div(x+1)$$

Dwijendra Rao

Numerade Educator

Problem 29

Perform the indicated divisions by synthetic division.
$$\left(x^{7}-128\right) \div(x-2)$$

Dwijendra Rao

Numerade Educator

Problem 30

Perform the indicated divisions by synthetic division.
$$\left(20 x^{4}+11 x^{3}-89 x^{2}+60 x-77\right) \div(x+2.75)$$

Carson Merrill

Numerade Educator

Problem 31

Perform the indicated divisions by synthetic division.
$$\left(2 x^{4}+x^{3}+3 x^{2}-1\right) \div(2 x-1)$$

Dwijendra Rao

Numerade Educator

Problem 32

Perform the indicated divisions by synthetic division.
$$\left(6 t^{4}+5 t^{3}-10 t+4\right) \div(3 t-2)$$

Dwijendra Rao

Numerade Educator

Problem 33

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.
$$2 x^{5}-x^{3}+3 x^{2}-4 ; \quad x+1$$

Dwijendra Rao

Numerade Educator

Problem 34

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.
$$t^{5}-3 t^{4}-t^{2}-6 ; t-3$$

Dwijendra Rao

Numerade Educator

Problem 35

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.
$$4 x^{3}-9 x^{2}+2 x-2 ; \quad x-\frac{1}{4}$$

Dwijendra Rao

Numerade Educator

Problem 36

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.
$$3 x^{3}-5 x^{2}+x+1 ; \quad x+\frac{1}{3}$$

Dwijendra Rao

Numerade Educator

Problem 37

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.
$$2 Z^{4}-Z^{3}-4 Z^{2}+1 ; \quad 2 Z-1$$

Dwijendra Rao

Numerade Educator

Problem 38

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.
$$6 x^{4}+5 x^{3}-x^{2}+6 x-2 ; \quad 3 x-1$$

Dwijendra Rao

Numerade Educator

Problem 39

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.
$$4 x^{4}+2 x^{3}-8 x^{2}+3 x+12 ; \quad 2 x+3$$

Dwijendra Rao

Numerade Educator

Problem 40

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.
$$3 x^{4}-2 x^{3}+x^{2}+15 x+4 ; \quad 3 x+4$$

Dwijendra Rao

Numerade Educator

Problem 41

Use synthetic division to determine whether or not the given numbers are zeros of the given functions.
$$x^{4}-5 x^{3}-15 x^{2}+5 x+14$$

Dwijendra Rao

Numerade Educator

Problem 42

Use synthetic division to determine whether or not the given numbers are zeros of the given functions.
$$r^{4}+5 r^{3}-18 r-8 ;-4$$

Dwijendra Rao

Numerade Educator

Problem 43

Use synthetic division to determine whether or not the given numbers are zeros of the given functions.
$$85 x^{3}+348 x^{2}-263 x+120 ; \quad-4.8$$

Dwijendra Rao

Numerade Educator

Problem 44

Use synthetic division to determine whether or not the given numbers are zeros of the given functions.
$$2 x^{3}+13 x^{2}+10 x-4 ; \quad \frac{1}{2}$$

Dwijendra Rao

Numerade Educator

Problem 45

Solve the given problems.
If $f(x)=2 x^{3}+3 x^{2}-19 x-4,$ and $f(x)=(x+4) g(x)$ find $g(x).$

Dwijendra Rao

Numerade Educator

Problem 46

Solve the given problems.
Using synthetic division, divide $a x^{2}+b x+c$ by $x+1.$

Dwijendra Rao

Numerade Educator

Problem 47

Solve the given problems.
By division, show that $2 x-1$ is a factor of $f(x)=4 x^{3}+8 x^{2}-x-2 .$ May we therefore conclude that $f(1)=0 ?$ Explain.

Dwijendra Rao

Numerade Educator

Problem 48

Solve the given problems.
By division, show that $x^{2}+2$ is a factor of $f(x)=3 x^{3}-x^{2}+6 x-2 .$ May we therefore conclude that $f(-2)=0 ?$ Explain.

Dwijendra Rao

Numerade Educator

Problem 49

Solve the given problems.
For what value of $k$ is $x-2$ a factor of $f(x)=2 x^{3}+k x^{2}-x+14 ?$

Dwijendra Rao

Numerade Educator

Problem 50

Solve the given problems.
For what value of $k$ is $x+1$ a factor of $f(x)=3 x^{4}+3 x^{3}+2 x^{2}+k x-4 ?$

Dwijendra Rao

Numerade Educator

Problem 51

Solve the given problems.
Use synthetic division: $\left(x^{3}-3 x^{2}+x-3\right) \div(x+j).$

Carson Merrill

Numerade Educator

Problem 52

Solve the given problems.
Use synthetic division: $$\left(2 x^{3}-7 x^{2}+10 x-6\right) \div[x-(1+j)].$$

Carson Merrill

Numerade Educator

Problem 53

Solve the given problems.
If $f(x)=-g(x),$ do the functions have the same zeros? Explain.

Dwijendra Rao

Numerade Educator

Problem 54

Solve the given problems.
Do the functions $f(x)$ and $f(-x)$ have the same zeros? Explain.

Dwijendra Rao

Numerade Educator

Problem 55

Solve the given problems.
If $f(x)=3 x^{3}-5 a x^{2}-3 a^{2} x+5 a^{3},$ find $f(x) \div(x+a).$

Dwijendra Rao

Numerade Educator

Problem 56

Solve the given problems.
The length of a rectangular box is $3 \mathrm{cm}$ longer than its width. If the volume as a function of the width is $f(w)=2 w^{3}+5 w^{2}-3 w$ find the height if the box.

Carson Merrill

Numerade Educator

Problem 57

Solve the given problems.
In finding the electric current in a certain circuit, it is necessary to factor the denominator of $\frac{2 s}{s^{3}+5 s^{2}+4 s+20}$. Is (a) $(s-2)$ or (b) $(s+5)$ a factor?

Dwijendra Rao

Numerade Educator

Problem 58

Solve the given problems.
In the theory of the motion of a sphere moving through a fluid, the function $f(r)=4 r^{3}-3 a r^{2}-a^{3}$ is used. Is (a) $r=a$ or (b) $r=2 a$ a zero of $f(r) ?$

Dwijendra Rao

Numerade Educator

Problem 59

Solve the given problems.
In finding the volume $V\left(\text { in } \mathrm{cm}^{3}$ ) of a certain gas in equilibrium with \right. a liquid, it is necessary to solve the equation $V^{3}-6 V^{2}+12 V=8$ Use synthetic division to determine if $V=2 \mathrm{cm}^{3}.$

Dwijendra Rao

Numerade Educator

Problem 60

Solve the given problems.
An architect is designing a window in the shape of a segment of a circle. An approximate formula for the area is $A=\frac{h^{3}}{2 w}+\frac{2 w h}{3}$ where $A$ is the area, $w$ is the width, and $h$ is the height of the segment. If the width is $1.500 \mathrm{m}$ and the area is $0.5417 \mathrm{m}^{2},$ use synthetic division to show that $h=0.500 \mathrm{m}.$

Carson Merrill

Numerade Educator