Question 1

Find the product. $$\left( - 3 a ^ { 2 } b \right) \left( - 3 a ^ { 3 } b ^ { 2 } \right) ( 6 a - 7 b )$$

Accepted Answer

A)  $21 a ^ { 5 } b ^ { 4 } - 18 a ^ { 6 } b ^ { 3 }$ 
B)  $54 a ^ { 5 } b ^ { 3 } + 42 a ^ { 5 } b ^ { 4 }$ 
C)  $54 a ^ { 6 } b ^ { 3 } - 63 a ^ { 5 } b ^ { 4 }$ 
D)  $63 a ^ { 5 } b ^ { 4 } - 54 a ^ { 6 } b ^ { 3 }$ 
E)  $18 a ^ { 6 } b ^ { 3 } - 21 a ^ { 5 } b ^ { 4 }$ 
A)  $21 a ^ { 5 } b ^ { 4 } - 18 a ^ { 6 } b ^ { 3 }$ 
B)  $54 a ^ { 5 } b ^ { 3 } + 42 a ^ { 5 } b ^ { 4 }$ 
C)  $54 a ^ { 6 } b ^ { 3 } - 63 a ^ { 5 } b ^ { 4 }$ 
D)  $63 a ^ { 5 } b ^ { 4 } - 54 a ^ { 6 } b ^ { 3 }$ 
E)  $18 a ^ { 6 } b ^ { 3 } - 21 a ^ { 5 } b ^ { 4 }$

Question 2

Factor the expression. Factor out any greatest common factors first. $4 x ^ { 3 } + 32$

Accepted Answer

A)  $4 ( x - 2 ) \left( x ^ { 2 } + 8 x + 4 \right)$ 
B)  $4 ( x + 4 ) \left( x ^ { 2 } + 4 x + 4 \right)$ 
C)  $2 ( x - 2 ) \left( x ^ { 2 } + 2 x + 4 \right)$ 
D)  $4 ( x + 2 ) \left( x ^ { 2 } - 2 x + 4 \right)$ 
E)  $2 ( x + 4 ) \left( x ^ { 2 } + 4 x - 16 \right)$ 
A)  $4 ( x - 2 ) \left( x ^ { 2 } + 8 x + 4 \right)$ 
B)  $4 ( x + 4 ) \left( x ^ { 2 } + 4 x + 4 \right)$ 
C)  $2 ( x - 2 ) \left( x ^ { 2 } + 2 x + 4 \right)$ 
D)  $4 ( x + 2 ) \left( x ^ { 2 } - 2 x + 4 \right)$ 
E)  $2 ( x + 4 ) \left( x ^ { 2 } + 4 x - 16 \right)$

Question 3

Find the quotient. $\frac { - 20 x ^ { 5 } y ^ { 9 } z ^ { 9 } } { x ^ { 2 } y z ^ { 3 } }$

Accepted Answer

A)  $- 20 x ^ { 7 } y ^ { 8 } z ^ { 6 }$ 
B)  $- 20 x ^ { 7 } y ^ { 9 } z ^ { 12 }$ 
C)  $- 20 x ^ { 7 } y ^ { 10 } z ^ { 12 }$ 
D)  $- 20 x ^ { 3 } y ^ { 8 } z ^ { 6 }$ 
E)  $- 20 x ^ { 3 } y ^ { 8 } z ^ { 12 }$ 
A)  $- 20 x ^ { 7 } y ^ { 8 } z ^ { 6 }$ 
B)  $- 20 x ^ { 7 } y ^ { 9 } z ^ { 12 }$ 
C)  $- 20 x ^ { 7 } y ^ { 10 } z ^ { 12 }$ 
D)  $- 20 x ^ { 3 } y ^ { 8 } z ^ { 6 }$ 
E)  $- 20 x ^ { 3 } y ^ { 8 } z ^ { 12 }$

Question 4

Find the product. $( 7 z + 1 ) ( 7 z - 1 )$

Accepted Answer

A)  $49 z ^ { 2 } - 1$ 
B)  $49 z ^ { 2 } - 14 z - 1$ 
C)  $49 z ^ { 2 } + 1$ 
D)  $14 z ^ { 2 } - 1$ 
E)  $14 z ^ { 2 } + 1$ 
A)  $49 z ^ { 2 } - 1$ 
B)  $49 z ^ { 2 } - 14 z - 1$ 
C)  $49 z ^ { 2 } + 1$ 
D)  $14 z ^ { 2 } - 1$ 
E)  $14 z ^ { 2 } + 1$

Question 5

Find (9 x + 9)3 . Don't forget that (9 x + 9)3 means (9 x + 9)(9 x + 9)(9 x + 9).

Accepted Answer

A) 729 x 3 + 729 x 2 + 2,187 x + 729 
B) 729 x 3 + 2,187 x 2 + 2,187 x + 81 
C) 729 x 3 + 2,187 x 2 + 729 x + 729 
D) 729 x 3 + 243 x 2 + 2,187 x + 729 
E) 729 x 3 + 2,187 x 2 + 2,187 x + 729 
A) 729 x 3 + 729 x 2 + 2,187 x + 729 
B) 729 x 3 + 2,187 x 2 + 2,187 x + 81 
C) 729 x 3 + 2,187 x 2 + 729 x + 729 
D) 729 x 3 + 243 x 2 + 2,187 x + 729 
E) 729 x 3 + 2,187 x 2 + 2,187 x + 729

Question 6

Use the rational root theorem and the factor theorem to help solve the equation. Be sure that the number of solutions for the equation agrees with property of number solutions of a polynomial equation, taking into account multiplicity of solutions. $2 x ^ { 4 } + x ^ { 3 } - 16 x ^ { 2 } - 3 x + 30 = 0$

Accepted Answer

A)  $\left\{ - \frac { 5 } { 2 } , 2 , \pm \sqrt { 3 } \right\}$ 
B)  $\left\{ - \frac { 3 } { 2 } , - 2 , \pm \sqrt { 3 } \right\}$ 
C)  $\left\{ - \frac { 5 } { 2 } , - 1 \pm \sqrt { 3 } \right\}$ 
D)  $\left\{ - \frac { 9 } { 2 } , 1 \pm \sqrt { 3 } \right\}$ 
E)  $\left\{ - \frac { 5 } { 2 } , - 2 , \pm \sqrt { 3 } \right\}$ 
A)  $\left\{ - \frac { 5 } { 2 } , 2 , \pm \sqrt { 3 } \right\}$ 
B)  $\left\{ - \frac { 3 } { 2 } , - 2 , \pm \sqrt { 3 } \right\}$ 
C)  $\left\{ - \frac { 5 } { 2 } , - 1 \pm \sqrt { 3 } \right\}$ 
D)  $\left\{ - \frac { 9 } { 2 } , 1 \pm \sqrt { 3 } \right\}$ 
E)  $\left\{ - \frac { 5 } { 2 } , - 2 , \pm \sqrt { 3 } \right\}$

Question 7

The perimeter of a rectangle is 22 inches and the area is 30 square inches. Find the length and width of the rectangle.

Accepted Answer

A) 5 and 12 
B) 10 and 12 
C) 5 and 6 
D) 10 and 6 
E) none of these 
A) 5 and 12 
B) 10 and 12 
C) 5 and 6 
D) 10 and 6 
E) none of these

Question 8

Find a polynomial that represents the volume of the rectangular solid, where a = 6 x , b = 3 x , c = 4 x .

Accepted Answer

A) V ( x )= 72 x 3 
B) V ( x )= 22 x 3 
C) V ( x )= 13 x 3 
D) V ( x )= 18 x 3 
E) V ( x )= 14 x 3 
A) V ( x )= 72 x 3 
B) V ( x )= 22 x 3 
C) V ( x )= 13 x 3 
D) V ( x )= 18 x 3 
E) V ( x )= 14 x 3

Question 9

Use the factoring method to solve the equation. $$6 x ^ { 2 } + 11 x + 4 = 0$$

Accepted Answer

A)  $- \frac { 4 } { 3 }$ , $- \frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 2 }$ , $\frac { 4 } { 3 }$ 
C)  $- \frac { 1 } { 2 }$ , $\frac { 4 } { 3 }$ 
D)  $- \frac { 4 } { 3 }$ , $\frac { 4 } { 3 }$ 
E)  $- \frac { 4 } { 3 }$ , $\frac { 1 } { 2 }$ 
A)  $- \frac { 4 } { 3 }$ , $- \frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 2 }$ , $\frac { 4 } { 3 }$ 
C)  $- \frac { 1 } { 2 }$ , $\frac { 4 } { 3 }$ 
D)  $- \frac { 4 } { 3 }$ , $\frac { 4 } { 3 }$ 
E)  $- \frac { 4 } { 3 }$ , $\frac { 1 } { 2 }$

Question 10

Find a polynomial that represents the total surface area of the right circular cylinder, and use that polynomial to determine the total surface area of the following right circular cylinder that has a base with a radius of 4. Use 3.14 for $\pi$ , and express the answers to the nearest tenth. Let h = 7.

Accepted Answer

A) A = 176.1 
B) A = 138.2 
C) A = 326.1 
D) A = 276.3 
E) A = 226.1 
A) A = 176.1 
B) A = 138.2 
C) A = 326.1 
D) A = 276.3 
E) A = 226.1

Question 11

Factor the expression. $$2 x ^ { 3 } z - 4 x ^ { 2 } z + 28 x z - 56 z$$

Accepted Answer

A)  $2 z ( x - 2 ) \left( x ^ { 2 } + 14 \right)$ 
B)  $2 z ( x + 2 ) \left( x ^ { 2 } - 14 \right)$ 
C)  $2 z ( x + 2 ) \left( x ^ { 2 } + 28 \right)$ 
D)  $2 z ( x + 2 ) \left( x ^ { 2 } + 14 \right)$ 
E)  $2 z ( x - 2 ) \left( x ^ { 2 } - 14 \right)$ 
A)  $2 z ( x - 2 ) \left( x ^ { 2 } + 14 \right)$ 
B)  $2 z ( x + 2 ) \left( x ^ { 2 } - 14 \right)$ 
C)  $2 z ( x + 2 ) \left( x ^ { 2 } + 28 \right)$ 
D)  $2 z ( x + 2 ) \left( x ^ { 2 } + 14 \right)$ 
E)  $2 z ( x - 2 ) \left( x ^ { 2 } - 14 \right)$

Question 12

Raise the monomial to the indicated power. $\left( 5 x y ^ { 5 } \right) ^ { 3 }$

Accepted Answer

A)  $125 x ^ { 3 } y ^ { 8 }$ 
B)  $125 x ^ { 4 } y ^ { 8 }$ 
C)  $125 x ^ { 3 } y ^ { 15 }$ 
D)  $15 x ^ { 4 } y ^ { 8 }$ 
E)  $15 x ^ { 3 } y ^ { 8 }$ 
A)  $125 x ^ { 3 } y ^ { 8 }$ 
B)  $125 x ^ { 4 } y ^ { 8 }$ 
C)  $125 x ^ { 3 } y ^ { 15 }$ 
D)  $15 x ^ { 4 } y ^ { 8 }$ 
E)  $15 x ^ { 3 } y ^ { 8 }$

Question 13

Find the indicated product. (3 x - 6) 3

Accepted Answer

A) 27 x  3 - 162 x  2 + 324 x - 6 
B) 27 x  3 - 162 x  2 - 54 x + 6 
C) 27 x  3 - 162 x  2 + 324 x - 216 
D) 27 x  3 + 162 x  2 - 324 x + 216 
E) 27 x  3 - 324 x  2 + 162 x - 216 
A) 27 x  3 - 162 x  2 + 324 x - 6 
B) 27 x  3 - 162 x  2 - 54 x + 6 
C) 27 x  3 - 162 x  2 + 324 x - 216 
D) 27 x  3 + 162 x  2 - 324 x + 216 
E) 27 x  3 - 324 x  2 + 162 x - 216

Question 14

Factor the composite number 21 into the product of prime numbers. For example: 30 = 2 * 3 * 5.

Accepted Answer

A)  $7 \cdot 3$ 
B)  $21 \cdot 1$ 
C)  $3 + 18$ 
D)  $19 + 2$ 
E)  $1 \cdot 21$ 
A)  $7 \cdot 3$ 
B)  $21 \cdot 1$ 
C)  $3 + 18$ 
D)  $19 + 2$ 
E)  $1 \cdot 21$

Question 15

Factor the polynomial. $2 ( x + y ) ^ { 2 } + ( x + y ) - 3$

Accepted Answer

A)  $( x + y + 3 ) ( 2 x + 2 y - 1 )$ 
B)  $( x + y + 1 ) ( 2 x + 2 y - 1 )$ 
C)  $( x + y + 3 ) ( x + y - 1 )$ 
D)  $( 2 x + 2 y + 1 ) ( x + y - 1 )$ 
E)  $( 2 x + 2 y + 3 ) ( x + y - 1 )$ 
A)  $( x + y + 3 ) ( 2 x + 2 y - 1 )$ 
B)  $( x + y + 1 ) ( 2 x + 2 y - 1 )$ 
C)  $( x + y + 3 ) ( x + y - 1 )$ 
D)  $( 2 x + 2 y + 1 ) ( x + y - 1 )$ 
E)  $( 2 x + 2 y + 3 ) ( x + y - 1 )$

Question 16

Find the product. $$\frac { 8 } { 3 } a ^ { 2 } b ( 6 a - 8 b )$$

Accepted Answer

A)  $\frac { 16 } { 3 } a ^ { 2 } b - \frac { 64 } { 3 } a ^ { 2 } b ^ { 2 }$ 
B)  $\frac { 16 } { 3 } a ^ { 3 } b + \frac { 64 } { 3 } a ^ { 2 } b ^ { 2 }$ 
C)  $16 a ^ { 3 } b - \frac { 64 } { 3 } a ^ { 2 } b ^ { 2 }$ 
D)  $16 a ^ { 3 } b - \frac { 56 } { 3 } a b ^ { 2 }$ 
E)  $16 a ^ { 2 } b + \frac { 64 } { 3 } a ^ { 2 } b ^ { 2 }$ 
A)  $\frac { 16 } { 3 } a ^ { 2 } b - \frac { 64 } { 3 } a ^ { 2 } b ^ { 2 }$ 
B)  $\frac { 16 } { 3 } a ^ { 3 } b + \frac { 64 } { 3 } a ^ { 2 } b ^ { 2 }$ 
C)  $16 a ^ { 3 } b - \frac { 64 } { 3 } a ^ { 2 } b ^ { 2 }$ 
D)  $16 a ^ { 3 } b - \frac { 56 } { 3 } a b ^ { 2 }$ 
E)  $16 a ^ { 2 } b + \frac { 64 } { 3 } a ^ { 2 } b ^ { 2 }$

Question 17

Factor the polynomial completely. $$16 - 25 y ^ { 2 }$$

Accepted Answer

A) (4 + y )(4 + 5 y ) 
B) (4 - 5 y )(4 + 5 y ) 
C) (16 - 5 y )(16 + 5 y ) 
D) (4 - 25 y )(4 + 25 y ) 
E) none of these 
A) (4 + y )(4 + 5 y ) 
B) (4 - 5 y )(4 + 5 y ) 
C) (16 - 5 y )(16 + 5 y ) 
D) (4 - 25 y )(4 + 25 y ) 
E) none of these

Question 18

Find the quotient. $\frac { - 20 a b c ^ { 7 } } { 20 b c }$

Accepted Answer

A)  $- a b c ^ { 8 }$ 
B)  $- a c ^ { 6 }$ 
C)  $- a b c ^ { 6 }$ 
D)  $- a c ^ { 8 }$ 
E)  $- a c ^ { 7 }$ 
A)  $- a b c ^ { 8 }$ 
B)  $- a c ^ { 6 }$ 
C)  $- a b c ^ { 6 }$ 
D)  $- a c ^ { 8 }$ 
E)  $- a c ^ { 7 }$

Question 19

The combined area of two squares is 104 square meters. The sides of the larger square are 5 times as long as the sides of the smaller square. Find the dimensions of each of the squares.

Accepted Answer

A) Dimensions of the smaller square are 1 by 1 meters, dimensions of the larger square are 2 by 2 meters. 
B) Dimensions of the smaller square are 2 by 2 meters, dimensions of the larger square are 10 by 10 meters. 
C) Dimensions of the smaller square are 3 by 3 meters, dimensions of the larger square are 15 by 15 meters. 
D) Dimensions of the smaller square are 7 by 7 meters, dimensions of the larger square are 35 by 35 meters. 
E) Dimensions of the smaller square are 1 by 1 meters, dimensions of the larger square are 5 by 5 meters. 
A) Dimensions of the smaller square are 1 by 1 meters, dimensions of the larger square are 2 by 2 meters. 
B) Dimensions of the smaller square are 2 by 2 meters, dimensions of the larger square are 10 by 10 meters. 
C) Dimensions of the smaller square are 3 by 3 meters, dimensions of the larger square are 15 by 15 meters. 
D) Dimensions of the smaller square are 7 by 7 meters, dimensions of the larger square are 35 by 35 meters. 
E) Dimensions of the smaller square are 1 by 1 meters, dimensions of the larger square are 5 by 5 meters.

Question 20

Factor the polynomial. $x ^ { 3 } + 8$

Accepted Answer

A)  $( x + 2 ) \left( x ^ { 2 } - 2 x + 4 \right)$ 
B)  $( x - 2 ) \left( x ^ { 2 } - 2 x + 4 \right)$ 
C)  $( x + 2 ) \left( x ^ { 2 } - 2 x - 4 \right)$ 
D)  $( x + 2 ) ^ { 3 }$ 
E)  $( x - 2 ) ^ { 3 }$ 
A)  $( x + 2 ) \left( x ^ { 2 } - 2 x + 4 \right)$ 
B)  $( x - 2 ) \left( x ^ { 2 } - 2 x + 4 \right)$ 
C)  $( x + 2 ) \left( x ^ { 2 } - 2 x - 4 \right)$ 
D)  $( x + 2 ) ^ { 3 }$ 
E)  $( x - 2 ) ^ { 3 }$

Find the product. $\left( - 3 a ^ { 2 } b \right) \left( - 3 a ^ { 3 } b ^ { 2 } \right) ( 6 a - 7 b )$

Factor the expression. Factor out any greatest common factors first. $4 x ^ { 3 } + 32$

Find the quotient. $\frac { - 20 x ^ { 5 } y ^ { 9 } z ^ { 9 } } { x ^ { 2 } y z ^ { 3 } }$

Find the product. $( 7 z + 1 ) ( 7 z - 1 )$

Find (9 x + 9)³. Don't forget that (9 x + 9)³ means (9 x + 9)(9 x + 9)(9 x + 9).

The perimeter of a rectangle is 22 inches and the area is 30 square inches. Find the length and width of the rectangle.

Find a polynomial that represents the volume of the rectangular solid, where a = 6 x , b = 3 x , c = 4 x .

Use the factoring method to solve the equation. $6 x ^ { 2 } + 11 x + 4 = 0$

Factor the expression. $2 x ^ { 3 } z - 4 x ^ { 2 } z + 28 x z - 56 z$

Raise the monomial to the indicated power. $\left( 5 x y ^ { 5 } \right) ^ { 3 }$

Find the indicated product. (3 x - 6)³

Factor the composite number 21 into the product of prime numbers. For example: 30 = 2 * 3 * 5.

Factor the polynomial. $2 ( x + y ) ^ { 2 } + ( x + y ) - 3$

Find the product. $\frac { 8 } { 3 } a ^ { 2 } b ( 6 a - 8 b )$

Factor the polynomial completely. $16 - 25 y ^ { 2 }$

Find the quotient. $\frac { - 20 a b c ^ { 7 } } { 20 b c }$

The combined area of two squares is 104 square meters. The sides of the larger square are 5 times as long as the sides of the smaller square. Find the dimensions of each of the squares.

Factor the polynomial. $x ^ { 3 } + 8$

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Exam 3: Polynomials

Find the product. (−3a2b)(−3a3b2)(6a−7b)\left( - 3 a ^ { 2 } b \right) \left( - 3 a ^ { 3 } b ^ { 2 } \right) ( 6 a - 7 b )(−3a2b)(−3a3b2)(6a−7b)

Factor the expression. Factor out any greatest common factors first. 4x3+324 x ^ { 3 } + 324x3+32

Find the quotient. −20x5y9z9x2yz3\frac { - 20 x ^ { 5 } y ^ { 9 } z ^ { 9 } } { x ^ { 2 } y z ^ { 3 } }x2yz3−20x5y9z9​

Find the product. (7z+1)(7z−1)( 7 z + 1 ) ( 7 z - 1 )(7z+1)(7z−1)

Find (9 x + 9)3 . Don't forget that (9 x + 9)3 means (9 x + 9)(9 x + 9)(9 x + 9).

The perimeter of a rectangle is 22 inches and the area is 30 square inches. Find the length and width of the rectangle.

Find a polynomial that represents the volume of the rectangular solid, where a = 6 x , b = 3 x , c = 4 x .

Use the factoring method to solve the equation. 6x2+11x+4=06 x ^ { 2 } + 11 x + 4 = 06x2+11x+4=0

Factor the expression. 2x3z−4x2z+28xz−56z2 x ^ { 3 } z - 4 x ^ { 2 } z + 28 x z - 56 z2x3z−4x2z+28xz−56z

Raise the monomial to the indicated power. (5xy5)3\left( 5 x y ^ { 5 } \right) ^ { 3 }(5xy5)3

Find the indicated product. (3 x - 6) 3

Factor the composite number 21 into the product of prime numbers. For example: 30 = 2 * 3 * 5.

Factor the polynomial. 2(x+y)2+(x+y)−32 ( x + y ) ^ { 2 } + ( x + y ) - 32(x+y)2+(x+y)−3

Find the product. 83a2b(6a−8b)\frac { 8 } { 3 } a ^ { 2 } b ( 6 a - 8 b )38​a2b(6a−8b)

Factor the polynomial completely. 16−25y216 - 25 y ^ { 2 }16−25y2

Find the quotient. −20abc720bc\frac { - 20 a b c ^ { 7 } } { 20 b c }20bc−20abc7​

The combined area of two squares is 104 square meters. The sides of the larger square are 5 times as long as the sides of the smaller square. Find the dimensions of each of the squares.

Factor the polynomial. x3+8x ^ { 3 } + 8x3+8

Basic Concepts and Properties

Equations, Inequalities, and Problem Solving

Rational Expressions

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Find the product. $\left( - 3 a ^ { 2 } b \right) \left( - 3 a ^ { 3 } b ^ { 2 } \right) ( 6 a - 7 b )$

Factor the expression. Factor out any greatest common factors first. $4 x ^ { 3 } + 32$

Find the quotient. $\frac { - 20 x ^ { 5 } y ^ { 9 } z ^ { 9 } } { x ^ { 2 } y z ^ { 3 } }$

Find the product. $( 7 z + 1 ) ( 7 z - 1 )$

Find (9 x + 9)³. Don't forget that (9 x + 9)³ means (9 x + 9)(9 x + 9)(9 x + 9).

Use the factoring method to solve the equation. $6 x ^ { 2 } + 11 x + 4 = 0$

Factor the expression. $2 x ^ { 3 } z - 4 x ^ { 2 } z + 28 x z - 56 z$

Raise the monomial to the indicated power. $\left( 5 x y ^ { 5 } \right) ^ { 3 }$

Find the indicated product. (3 x - 6)³

Factor the polynomial. $2 ( x + y ) ^ { 2 } + ( x + y ) - 3$

Find the product. $\frac { 8 } { 3 } a ^ { 2 } b ( 6 a - 8 b )$

Factor the polynomial completely. $16 - 25 y ^ { 2 }$

Find the quotient. $\frac { - 20 a b c ^ { 7 } } { 20 b c }$

Factor the polynomial. $x ^ { 3 } + 8$