Question 1

Simplify the rational expression.
-$\frac { a ^ { 2 } - a b + 12 a - 12 b } { a + 12 }$

Accepted Answer

A)  $\frac { a ^ { 2 } - a b + 12 a - 12 b } { a + 12 }$ 
B)  $a - 2 b + 1$ 
C) a - b 
D)  $\frac { 1 } { a + 12 }$ 
A)  $\frac { a ^ { 2 } - a b + 12 a - 12 b } { a + 12 }$ 
B)  $a - 2 b + 1$ 
C) a - b 
D)  $\frac { 1 } { a + 12 }$ C

Question 2

Multiply.
-$\frac { 1 } { x + 8 } \cdot \frac { x ^ { 2 } + 18 x + 80 } { x + 10 }$

Accepted Answer

A) 1 
B)  $\frac { x - 10 } { x + 8 }$ 
C) x + 8 
D)  $\frac { x - 8 } { x + 10 }$ 
A) 1 
B)  $\frac { x - 10 } { x + 8 }$ 
C) x + 8 
D)  $\frac { x - 8 } { x + 10 }$ A

Question 3

Multiply.
-$\frac { 3 m ^ { 2 } + 6 m n + 3 n ^ { 2 } } { m ^ { 2 } - n ^ { 2 } } \cdot \frac { n - m } { 8 m + 8 n }$

Accepted Answer

A)  $\frac { 3 } { 8 ( m + n ) }$ 
B)  $\frac { 3 } { 8 }$ 
C)  $- \frac { 3 } { 8 }$ 
D) -3 
A)  $\frac { 3 } { 8 ( m + n ) }$ 
B)  $\frac { 3 } { 8 }$ 
C)  $- \frac { 3 } { 8 }$ 
D) -3 C

Question 4

Simplify the rational expression.
-$\frac { ( x + 3 ) ( x - 8 ) + ( x + 3 ) ( x + 7 ) } { 3 ( x + 3 ) }$

Accepted Answer

A)  $\frac { 2 x - 1 } { 3 }$ 
B)  $- \frac { 1 } { 3 }$ 
C)  $2 x - 1$ 
D)  $\frac { x - 1 } { 3 }$ 
A)  $\frac { 2 x - 1 } { 3 }$ 
B)  $- \frac { 1 } { 3 }$ 
C)  $2 x - 1$ 
D)  $\frac { x - 1 } { 3 }$

Question 5

Simplify the rational expression.
-$\frac { y ^ { 8 } - y ^ { 6 } } { y ^ { 2 } - 1 }$

Accepted Answer

A)  $y ^ { 2 } - 1$ 
B)  $y ^ { 6 }$ 
C)  $y ^ { 2 }$ 
D) y 
A)  $y ^ { 2 } - 1$ 
B)  $y ^ { 6 }$ 
C)  $y ^ { 2 }$ 
D) y

Question 6

Multiply.
-$\frac { 9 x ^ { 4 } - 72 x } { 3 x ^ { 2 } - 12 } \cdot \frac { x ^ { 2 } + x - 2 } { 4 x ^ { 3 } + 8 x ^ { 2 } + 16 x }$

Accepted Answer

A)  $\frac { 3 ( x - 1 ) } { 4 }$ 
B)  $\frac { 3 x ( x - 1 ) ( x - 2 ) ^ { 2 } } { 4 ( x + 2 ) ^ { 2 } }$ 
C)  $\frac { 3 x ( x + 1 ) } { 4 }$ 
D)  $\frac { 3 x ( x - 1 ) } { 4 }$ 
A)  $\frac { 3 ( x - 1 ) } { 4 }$ 
B)  $\frac { 3 x ( x - 1 ) ( x - 2 ) ^ { 2 } } { 4 ( x + 2 ) ^ { 2 } }$ 
C)  $\frac { 3 x ( x + 1 ) } { 4 }$ 
D)  $\frac { 3 x ( x - 1 ) } { 4 }$

Question 7

Simplify the rational expression.
-$\frac { 125 x ^ { 3 } + 27 y ^ { 3 } } { 5 x + 3 y }$

Accepted Answer

A)  $5 x ^ { 2 } + x y + 3 y ^ { 2 }$ 
B)  $25 x ^ { 2 } + x y + 9 y ^ { 2 }$ 
C)  $25 x ^ { 2 } + 15 x y + 9 y ^ { 2 }$ 
D)  $25 x ^ { 2 } - 15 x y + 9 y ^ { 2 }$ 
A)  $5 x ^ { 2 } + x y + 3 y ^ { 2 }$ 
B)  $25 x ^ { 2 } + x y + 9 y ^ { 2 }$ 
C)  $25 x ^ { 2 } + 15 x y + 9 y ^ { 2 }$ 
D)  $25 x ^ { 2 } - 15 x y + 9 y ^ { 2 }$

Question 8

Simplify the rational expression.
-$\frac { c ^ { 3 } - d ^ { 3 } } { c ^ { 2 } - d ^ { 2 } }$

Accepted Answer

A)  $\frac { c ^ { 2 } - c d + d ^ { 2 } } { c + d }$ 
B) c+ d 
C)  $\frac { c ^ { 2 } + c d + d ^ { 2 } } { c + d }$ 
D)  $\frac { c ^ { 2 } - c d + d ^ { 2 } } { c - d }$ 
A)  $\frac { c ^ { 2 } - c d + d ^ { 2 } } { c + d }$ 
B) c+ d 
C)  $\frac { c ^ { 2 } + c d + d ^ { 2 } } { c + d }$ 
D)  $\frac { c ^ { 2 } - c d + d ^ { 2 } } { c - d }$

Question 9

Multiply.
-$\frac { 3 x y ^ { 5 } } { 5 x ^ { 6 } y ^ { 3 } } \cdot \frac { - 20 x ^ { 5 } y ^ { 8 } } { 9 x ^ { 7 } y ^ { 12 } }$

Accepted Answer

A)  $- \frac { 27 y ^ { 6 } } { 100 x ^ { 3 } }$ 
B)  $- \frac { 4 } { 3 x ^ { 6 } y ^ { 3 } }$ 
C)  $- \frac { 4 } { 3 x ^ { 7 } y ^ { 2 } }$ 
D)  $- \frac { 4 } { 3 x ^ { 8 } y ^ { 2 } }$ 
A)  $- \frac { 27 y ^ { 6 } } { 100 x ^ { 3 } }$ 
B)  $- \frac { 4 } { 3 x ^ { 6 } y ^ { 3 } }$ 
C)  $- \frac { 4 } { 3 x ^ { 7 } y ^ { 2 } }$ 
D)  $- \frac { 4 } { 3 x ^ { 8 } y ^ { 2 } }$

Question 10

Add or subtract.
-$\frac { 8 x - 8 } { x + 4 } + 1$

Accepted Answer

A)  $\frac { 8 x + 4 } { x + 4 }$ 
B)  $\frac { 7 x - 4 } { x + 4 }$ 
C)  $\frac { 9 x - 4 } { x + 4 }$ 
D)  $\frac { 8 x - 4 } { x + 4 }$ 
A)  $\frac { 8 x + 4 } { x + 4 }$ 
B)  $\frac { 7 x - 4 } { x + 4 }$ 
C)  $\frac { 9 x - 4 } { x + 4 }$ 
D)  $\frac { 8 x - 4 } { x + 4 }$

Question 11

Multiply.
-$\frac { x ^ { 2 } + 12 x + 32 } { x ^ { 2 } + 13 x + 40 } \cdot \frac { x ^ { 2 } + 7 x + 10 } { x ^ { 2 } + 6 x + 8 }$

Accepted Answer

A)  $\frac { x + 4 } { x + 5 }$ 
B)  $\frac { x + 5 } { x + 2 }$ 
C) 1 
D)  $\frac { 1 } { x + 2 }$ 
A)  $\frac { x + 4 } { x + 5 }$ 
B)  $\frac { x + 5 } { x + 2 }$ 
C) 1 
D)  $\frac { 1 } { x + 2 }$

Question 12

Find the least common denominator.
-$\frac { 7 } { 16 ( x + 6 ) } - \frac { 8 } { x }$

Accepted Answer

A) 16x(x + 6) 
B)  $16 \left( x ^ { 2 } + 6 \right)$ 
C)  $16 x ^ { 2 } + 6$ 
D)  $16 x + 6$ 
A) 16x(x + 6) 
B)  $16 \left( x ^ { 2 } + 6 \right)$ 
C)  $16 x ^ { 2 } + 6$ 
D)  $16 x + 6$

Question 13

Determine the domain of the function.
-$g ( x ) = \frac { x + 6 } { x ^ { 2 } + 49 }$

Accepted Answer

A)  $\{ x \mid x \neq - 7 \text { and } x \neq 7 \}$ 
B)  $\{ x \mid x \text { is a real number } \}$ 
C)  $\{ x \mid x \neq - 7 , x \neq 7 \text {, and } x \neq - 6 \}$ 
D)  $\{ x \mid x \neq 0 \text { and } x \neq - 49 \}$ 
A)  $\{ x \mid x \neq - 7 \text { and } x \neq 7 \}$ 
B)  $\{ x \mid x \text { is a real number } \}$ 
C)  $\{ x \mid x \neq - 7 , x \neq 7 \text {, and } x \neq - 6 \}$ 
D)  $\{ x \mid x \neq 0 \text { and } x \neq - 49 \}$

Question 14

Solve the problem.
-Consider the triangle below. If its area is $x ^ { 2 } + 13 x y + 12 y ^ { 2 }$ and its base is x + 12y, find its height, h. Use area = $\frac { 1 } { 2 }$ (base)(height).

Accepted Answer

A) x + y 
B)  $\frac { x + y } { 2 }$ 
C) 2(x + 12y) 
D) 2(x + y) 
A) x + y 
B)  $\frac { x + y } { 2 }$ 
C) 2(x + 12y) 
D) 2(x + y)

Question 15

Simplify the rational expression.
-$\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 6 x + 18 x ^ { 2 } }$

Accepted Answer

A)  $\frac { 7 + 21 x ^ { 3 } } { 6 x + 18 }$ 
B)  $\frac { 7 x } { 6 }$ 
C)  $\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 6 x + 18 x ^ { 2 } }$ 
D)  $\frac { 7 } { 6 }$ 
A)  $\frac { 7 + 21 x ^ { 3 } } { 6 x + 18 }$ 
B)  $\frac { 7 x } { 6 }$ 
C)  $\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 6 x + 18 x ^ { 2 } }$ 
D)  $\frac { 7 } { 6 }$

Question 16

Determine the domain of the function.
-$f ( x ) = \frac { x ^ { 3 } + 8 x ^ { 4 } } { x ^ { 2 } + 25 }$

Accepted Answer

A)  $\left\{ x \mid x \neq 0 \text { and } x \neq - \frac { 1 } { 8 } \right\}$ 
B)  $\{ x \mid x \neq - 5 \}$ 
C)  $\{ x \mid x \neq - 25 \}$ 
D)  $\{ \mathrm { x } \mid \mathrm { x } \text { is a real number } \}$ 
A)  $\left\{ x \mid x \neq 0 \text { and } x \neq - \frac { 1 } { 8 } \right\}$ 
B)  $\{ x \mid x \neq - 5 \}$ 
C)  $\{ x \mid x \neq - 25 \}$ 
D)  $\{ \mathrm { x } \mid \mathrm { x } \text { is a real number } \}$

Question 17

Add or subtract.
-$\frac { 2 } { 15 a ^ { 3 } b } - \frac { 1 } { 18 a b ^ { 2 } }$

Accepted Answer

A)  $\frac { 12 b - 1 a ^ { 2 } } { 90 a ^ { 3 } b ^ { 2 } }$ 
B)  $\frac { 12 b - 5 a ^ { 2 } } { 90 a ^ { 3 } b ^ { 2 } }$ 
C)  $\frac { 12 b - 5 a ^ { 2 } } { 90 a ^ { 2 } b ^ { 3 } }$ 
D)  $\frac { 12 b + 5 a ^ { 2 } } { 90 a ^ { 3 } b ^ { 2 } }$ 
A)  $\frac { 12 b - 1 a ^ { 2 } } { 90 a ^ { 3 } b ^ { 2 } }$ 
B)  $\frac { 12 b - 5 a ^ { 2 } } { 90 a ^ { 3 } b ^ { 2 } }$ 
C)  $\frac { 12 b - 5 a ^ { 2 } } { 90 a ^ { 2 } b ^ { 3 } }$ 
D)  $\frac { 12 b + 5 a ^ { 2 } } { 90 a ^ { 3 } b ^ { 2 } }$

Question 18

Add or subtract.
-$\frac { 4 x ^ { 2 } } { x - 1 } + \frac { - 4 x } { x - 1 }$

Accepted Answer

A) 0 
B) 4x 
C)  $\frac { 4 x } { x - 1 }$ 
D)  $\frac { 4 x ( x + 1 ) } { x - 1 }$ 
A) 0 
B) 4x 
C)  $\frac { 4 x } { x - 1 }$ 
D)  $\frac { 4 x ( x + 1 ) } { x - 1 }$

Question 19

Determine the domain of the function.
-$f ( x ) = \frac { x + 3 } { x ^ { 2 } - 1 }$

Accepted Answer

A)  $\{ x \mid x \neq - 1 \text { and } x \neq 1 \}$ 
B)  $\{ x \mid x \neq 0 \text { and } x \neq 1 \}$ 
C)  $\{ x \mid x \text { is a real number } \}$ 
D)  $\{ x \mid x \neq - 1 , x \neq 1 \text {, and } x \neq - 3 \}$ 
A)  $\{ x \mid x \neq - 1 \text { and } x \neq 1 \}$ 
B)  $\{ x \mid x \neq 0 \text { and } x \neq 1 \}$ 
C)  $\{ x \mid x \text { is a real number } \}$ 
D)  $\{ x \mid x \neq - 1 , x \neq 1 \text {, and } x \neq - 3 \}$

Question 20

Find the least common denominator.
-$\frac { 7 x + 13 } { 14 x - 19 } - \frac { 17 x } { 14 x + 19 }$

Accepted Answer

A)  $( 14 x - 19 ) \text { or } ( 14 x + 19 )$ 
B) 17x(7x + 13)(14x - 19)(14x + 19) 
C)  $( 14 x - 19 ) ( 14 x + 19 )$ 
D) 17x(7x + 13) 
A)  $( 14 x - 19 ) \text { or } ( 14 x + 19 )$ 
B) 17x(7x + 13)(14x - 19)(14x + 19) 
C)  $( 14 x - 19 ) ( 14 x + 19 )$ 
D) 17x(7x + 13)

Simplify the rational expression. - $\frac { a ^ { 2 } - a b + 12 a - 12 b } { a + 12 }$

Multiply. - $\frac { 1 } { x + 8 } \cdot \frac { x ^ { 2 } + 18 x + 80 } { x + 10 }$

Multiply. - $\frac { 3 m ^ { 2 } + 6 m n + 3 n ^ { 2 } } { m ^ { 2 } - n ^ { 2 } } \cdot \frac { n - m } { 8 m + 8 n }$

Simplify the rational expression. - $\frac { ( x + 3 ) ( x - 8 ) + ( x + 3 ) ( x + 7 ) } { 3 ( x + 3 ) }$

Simplify the rational expression. - $\frac { y ^ { 8 } - y ^ { 6 } } { y ^ { 2 } - 1 }$

Multiply. - $\frac { 9 x ^ { 4 } - 72 x } { 3 x ^ { 2 } - 12 } \cdot \frac { x ^ { 2 } + x - 2 } { 4 x ^ { 3 } + 8 x ^ { 2 } + 16 x }$

Simplify the rational expression. - $\frac { 125 x ^ { 3 } + 27 y ^ { 3 } } { 5 x + 3 y }$

Simplify the rational expression. - $\frac { c ^ { 3 } - d ^ { 3 } } { c ^ { 2 } - d ^ { 2 } }$

Multiply. - $\frac { 3 x y ^ { 5 } } { 5 x ^ { 6 } y ^ { 3 } } \cdot \frac { - 20 x ^ { 5 } y ^ { 8 } } { 9 x ^ { 7 } y ^ { 12 } }$

Add or subtract. - $\frac { 8 x - 8 } { x + 4 } + 1$

Multiply. - $\frac { x ^ { 2 } + 12 x + 32 } { x ^ { 2 } + 13 x + 40 } \cdot \frac { x ^ { 2 } + 7 x + 10 } { x ^ { 2 } + 6 x + 8 }$

Find the least common denominator. - $\frac { 7 } { 16 ( x + 6 ) } - \frac { 8 } { x }$

Determine the domain of the function. - $g ( x ) = \frac { x + 6 } { x ^ { 2 } + 49 }$

Simplify the rational expression. - $\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 6 x + 18 x ^ { 2 } }$

Determine the domain of the function. - $f ( x ) = \frac { x ^ { 3 } + 8 x ^ { 4 } } { x ^ { 2 } + 25 }$

Add or subtract. - $\frac { 2 } { 15 a ^ { 3 } b } - \frac { 1 } { 18 a b ^ { 2 } }$

Add or subtract. - $\frac { 4 x ^ { 2 } } { x - 1 } + \frac { - 4 x } { x - 1 }$

Determine the domain of the function. - $f ( x ) = \frac { x + 3 } { x ^ { 2 } - 1 }$

Find the least common denominator. - $\frac { 7 x + 13 } { 14 x - 19 } - \frac { 17 x } { 14 x + 19 }$

Basic Concepts

Equations and Inequalities

Graphs and Functions

Systems of Equations and Inequalities

Polynomials and Polynomial Functions

Roots, Radicals, and Complex Numbers

Quadratic Functions

Exponential and Logarithmic Functions

Characteristics of Functions and Their Graphs

Conic Sections

Sequences, Series, and Probability

Filters

Exam 6: Rational Expressions and Equations

Simplify the rational expression. - a2−ab+12a−12ba+12\frac { a ^ { 2 } - a b + 12 a - 12 b } { a + 12 }a+12a2−ab+12a−12b​

Multiply. - 1x+8⋅x2+18x+80x+10\frac { 1 } { x + 8 } \cdot \frac { x ^ { 2 } + 18 x + 80 } { x + 10 }x+81​⋅x+10x2+18x+80​

Multiply. - 3m2+6mn+3n2m2−n2⋅n−m8m+8n\frac { 3 m ^ { 2 } + 6 m n + 3 n ^ { 2 } } { m ^ { 2 } - n ^ { 2 } } \cdot \frac { n - m } { 8 m + 8 n }m2−n23m2+6mn+3n2​⋅8m+8nn−m​

Simplify the rational expression. - (x+3)(x−8)+(x+3)(x+7)3(x+3)\frac { ( x + 3 ) ( x - 8 ) + ( x + 3 ) ( x + 7 ) } { 3 ( x + 3 ) }3(x+3)(x+3)(x−8)+(x+3)(x+7)​

Simplify the rational expression. - y8−y6y2−1\frac { y ^ { 8 } - y ^ { 6 } } { y ^ { 2 } - 1 }y2−1y8−y6​

Multiply. - 9x4−72x3x2−12⋅x2+x−24x3+8x2+16x\frac { 9 x ^ { 4 } - 72 x } { 3 x ^ { 2 } - 12 } \cdot \frac { x ^ { 2 } + x - 2 } { 4 x ^ { 3 } + 8 x ^ { 2 } + 16 x }3x2−129x4−72x​⋅4x3+8x2+16xx2+x−2​

Simplify the rational expression. - 125x3+27y35x+3y\frac { 125 x ^ { 3 } + 27 y ^ { 3 } } { 5 x + 3 y }5x+3y125x3+27y3​

Simplify the rational expression. - c3−d3c2−d2\frac { c ^ { 3 } - d ^ { 3 } } { c ^ { 2 } - d ^ { 2 } }c2−d2c3−d3​

Multiply. - 3xy55x6y3⋅−20x5y89x7y12\frac { 3 x y ^ { 5 } } { 5 x ^ { 6 } y ^ { 3 } } \cdot \frac { - 20 x ^ { 5 } y ^ { 8 } } { 9 x ^ { 7 } y ^ { 12 } }5x6y33xy5​⋅9x7y12−20x5y8​

Add or subtract. - 8x−8x+4+1\frac { 8 x - 8 } { x + 4 } + 1x+48x−8​+1

Multiply. - x2+12x+32x2+13x+40⋅x2+7x+10x2+6x+8\frac { x ^ { 2 } + 12 x + 32 } { x ^ { 2 } + 13 x + 40 } \cdot \frac { x ^ { 2 } + 7 x + 10 } { x ^ { 2 } + 6 x + 8 }x2+13x+40x2+12x+32​⋅x2+6x+8x2+7x+10​

Find the least common denominator. - 716(x+6)−8x\frac { 7 } { 16 ( x + 6 ) } - \frac { 8 } { x }16(x+6)7​−x8​

Determine the domain of the function. - g(x)=x+6x2+49g ( x ) = \frac { x + 6 } { x ^ { 2 } + 49 }g(x)=x2+49x+6​

Solve the problem. -Consider the triangle below. If its area is x2+13xy+12y2x ^ { 2 } + 13 x y + 12 y ^ { 2 }x2+13xy+12y2 and its base is x + 12y, find its height, h. Use area = 12\frac { 1 } { 2 }21​ (base)(height).

Simplify the rational expression. - 7x2+21x36x+18x2\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 6 x + 18 x ^ { 2 } }6x+18x27x2+21x3​

Determine the domain of the function. - f(x)=x3+8x4x2+25f ( x ) = \frac { x ^ { 3 } + 8 x ^ { 4 } } { x ^ { 2 } + 25 }f(x)=x2+25x3+8x4​

Add or subtract. - 215a3b−118ab2\frac { 2 } { 15 a ^ { 3 } b } - \frac { 1 } { 18 a b ^ { 2 } }15a3b2​−18ab21​

Add or subtract. - 4x2x−1+−4xx−1\frac { 4 x ^ { 2 } } { x - 1 } + \frac { - 4 x } { x - 1 }x−14x2​+x−1−4x​

Determine the domain of the function. - f(x)=x+3x2−1f ( x ) = \frac { x + 3 } { x ^ { 2 } - 1 }f(x)=x2−1x+3​

Find the least common denominator. - 7x+1314x−19−17x14x+19\frac { 7 x + 13 } { 14 x - 19 } - \frac { 17 x } { 14 x + 19 }14x−197x+13​−14x+1917x​