Question 1

Simplify the rational expression.
-$\frac { y ^ { 2 } + 7 y + 10 } { y ^ { 2 } + 11 y + 30 }$

Accepted Answer

A)  $\frac { 7 y + 1 } { 11 y + 3 }$ 
B)  $- \frac { y ^ { 2 } + 7 y + 10 } { y ^ { 2 } + 11 y + 30 }$ 
C)  $\frac { 7 y + 10 } { 11 y + 30 }$ 
D)  $\frac { y + 2 } { y + 6 }$ 
A)  $\frac { 7 y + 1 } { 11 y + 3 }$ 
B)  $- \frac { y ^ { 2 } + 7 y + 10 } { y ^ { 2 } + 11 y + 30 }$ 
C)  $\frac { 7 y + 10 } { 11 y + 30 }$ 
D)  $\frac { y + 2 } { y + 6 }$

Question 2

Divide.
-$\frac { x ^ { 2 } - 10 x + 25 } { 7 x - 35 } \div \frac { 2 x - 10 } { 14 }$

Accepted Answer

A) 14 
B)  $\frac { ( x - 5 ) ^ { 2 } } { 49 }$ 
C) 1 
D)  $\frac { x ^ { 2 } - 10 x + 25 } { ( x - 5 ) ^ { 2 } }$ 
A) 14 
B)  $\frac { ( x - 5 ) ^ { 2 } } { 49 }$ 
C) 1 
D)  $\frac { x ^ { 2 } - 10 x + 25 } { ( x - 5 ) ^ { 2 } }$

Question 3

Divide.
-$\frac { 27 x - 27 } { 5 } \div \frac { 9 x - 9 } { 10 }$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 2 ( 27 x - 27 ) } { 9 x - 9 }$ 
C)  $\frac { 243 ( x - 1 ) ^ { 2 } } { 50 }$ 
D) 6 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 2 ( 27 x - 27 ) } { 9 x - 9 }$ 
C)  $\frac { 243 ( x - 1 ) ^ { 2 } } { 50 }$ 
D) 6

Question 4

Solve the problem.
-A parallelogram has areaas area $\frac { x ^ { 2 } + 9 x + 8 } { x } \text { and height } \frac { x ^ { 5 } } { x + 1 }$ . Find the length of its base, by dividing its area by its height.

Accepted Answer

A)  $\frac { x + 8 } { x ^ { 4 } }$ 
B)  $x ^ { 4 } ( x + 8 )$ 
C)  $\frac { ( x + 8 ) ( x + 1 ) ^ { 2 } } { x ^ { 4 } }$ 
D)  $\frac { ( x + 8 ) ( x + 1 ) ^ { 2 } } { x ^ { 6 } }$ 
A)  $\frac { x + 8 } { x ^ { 4 } }$ 
B)  $x ^ { 4 } ( x + 8 )$ 
C)  $\frac { ( x + 8 ) ( x + 1 ) ^ { 2 } } { x ^ { 4 } }$ 
D)  $\frac { ( x + 8 ) ( x + 1 ) ^ { 2 } } { x ^ { 6 } }$

Question 5

Multiply.
-$\frac { 4 x } { 8 x + 4 } \cdot \frac { 6 x + 3 } { 5 }$

Accepted Answer

A)  $\frac { 3 } { 5 }$ 
B)  $\frac { x } { 5 }$ 
C)  $\frac { 3 x } { 5 }$ 
D)  $\frac { 3 x } { 20 }$ 
A)  $\frac { 3 } { 5 }$ 
B)  $\frac { x } { 5 }$ 
C)  $\frac { 3 x } { 5 }$ 
D)  $\frac { 3 x } { 20 }$

Question 6

Find the least common denominator.
-$\frac { 9 } { 4 a + 8 } - \frac { 6 } { a ^ { 2 } + 2 a }$

Accepted Answer

A) 4a(a + 2) 
B)  $4 a ^ { 2 } + 8$ 
C)  $4 \mathrm { a } ^ { 2 } + 2$ 
D)  $4 a + 2$ 
A) 4a(a + 2) 
B)  $4 a ^ { 2 } + 8$ 
C)  $4 \mathrm { a } ^ { 2 } + 2$ 
D)  $4 a + 2$

Question 7

Add or subtract.
-$\frac { 5 y } { x ^ { 2 } - y ^ { 2 } } + \frac { 5 x } { y ^ { 2 } - x ^ { 2 } }$

Accepted Answer

A)  $\frac { 5 } { x - y }$ 
B)  $- \frac { 5 } { x + y }$ 
C)  $\frac { 5 } { x + y }$ 
D)  $\frac { y - x } { x ^ { 2 } - y ^ { 2 } }$ 
A)  $\frac { 5 } { x - y }$ 
B)  $- \frac { 5 } { x + y }$ 
C)  $\frac { 5 } { x + y }$ 
D)  $\frac { y - x } { x ^ { 2 } - y ^ { 2 } }$

Question 8

Simplify the rational expression.
-$\frac { 4 x - 14 x ^ { 2 } } { 2 x }$

Accepted Answer

A) 0 
B) 1 
C)  $2 - 7 x$ 
D)  $2 + 7 x$ 
A) 0 
B) 1 
C)  $2 - 7 x$ 
D)  $2 + 7 x$

Question 9

Add or subtract.
-$\frac { 4 x ^ { 2 } + 3 x + 3 } { x ^ { 2 } + 13 x + 40 } - \frac { 3 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } + 13 x + 40 }$

Accepted Answer

A)  $\frac { x + 1 } { x - 8 }$ 
B)  $\frac { x - 1 } { x + 8 }$ 
C)  $\frac { x + 1 } { x + 8 }$ 
D)  $\frac { x ^ { 2 } + 6 x + 5 } { x ^ { 2 } + 13 x + 40 }$ 
A)  $\frac { x + 1 } { x - 8 }$ 
B)  $\frac { x - 1 } { x + 8 }$ 
C)  $\frac { x + 1 } { x + 8 }$ 
D)  $\frac { x ^ { 2 } + 6 x + 5 } { x ^ { 2 } + 13 x + 40 }$

Question 10

Simplify the rational expression.
-$\frac { w ^ { 3 } - 5 w ^ { 2 } - 2 w + 10 } { w ^ { 2 } - 10 w + 25 }$

Accepted Answer

A)  $\frac { w + 5 } { w ^ { 2 } + 2 }$ 
B)  $\frac { w ^ { 2 } - 2 } { w - 5 }$ 
C)  $\frac { w ^ { 2 } + 2 } { w + 5 }$ 
D)  $\frac { w ^ { 2 } - 5 } { w - 2 }$ 
A)  $\frac { w + 5 } { w ^ { 2 } + 2 }$ 
B)  $\frac { w ^ { 2 } - 2 } { w - 5 }$ 
C)  $\frac { w ^ { 2 } + 2 } { w + 5 }$ 
D)  $\frac { w ^ { 2 } - 5 } { w - 2 }$

Question 11

Multiply.
-$\frac { 4 x + y } { - 4 x - 5 y } \cdot \frac { - 4 x ^ { 2 } - 9 x y - 5 y ^ { 2 } } { 4 x ^ { 2 } + 5 x y + y ^ { 2 } }$

Accepted Answer

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

Question 12

Determine the domain of the function.
-$r ( x ) = \frac { x ^ { 2 } + x - 20 } { x ^ { 2 } - 14 x + 48 }$

Accepted Answer

A)  $\{ x \mid x \neq 5 \text { and } x \neq - 4 \}$ 
B)  $\{ x \mid x \neq 6 \text { and } x \neq 8 \}$ 
C)  $\{ x \mid x \neq - 6 \text { and } x \neq - 8 \}$ 
D)  $\{ x \mid x \neq - 5 \text { and } x \neq 4 \}$ 
A)  $\{ x \mid x \neq 5 \text { and } x \neq - 4 \}$ 
B)  $\{ x \mid x \neq 6 \text { and } x \neq 8 \}$ 
C)  $\{ x \mid x \neq - 6 \text { and } x \neq - 8 \}$ 
D)  $\{ x \mid x \neq - 5 \text { and } x \neq 4 \}$

Question 13

Simplify the rational expression.
-$\frac { y ^ { 3 } - 64 } { y - 4 }$

Accepted Answer

A)  $y ^ { 2 } + 4 y + 16$ 
B)  $y ^ { 2 } - 16$ 
C)  $\frac { 1 } { y - 4 }$ 
D)  $\frac { y ^ { 3 } - 64 } { y - 4 }$ 
A)  $y ^ { 2 } + 4 y + 16$ 
B)  $y ^ { 2 } - 16$ 
C)  $\frac { 1 } { y - 4 }$ 
D)  $\frac { y ^ { 3 } - 64 } { y - 4 }$

Question 14

Determine the domain of the function.
-$f ( x ) = \frac { 2 x ^ { 2 } - 4 } { 3 x ^ { 2 } + 6 x - 45 }$

Accepted Answer

A)  $\{ x \mid x \neq 3 , x \neq - 3 \text {, and } x \neq - 5 \}$ 
B)  $\{ x \mid x \neq 3 \text { and } x \neq - 5 \}$ 
C)  $\{ x \mid x \neq - 3 \text { and } x \neq 5 \}$ 
D)  $\{ x \mid x \text { is a real number } \}$ 
A)  $\{ x \mid x \neq 3 , x \neq - 3 \text {, and } x \neq - 5 \}$ 
B)  $\{ x \mid x \neq 3 \text { and } x \neq - 5 \}$ 
C)  $\{ x \mid x \neq - 3 \text { and } x \neq 5 \}$ 
D)  $\{ x \mid x \text { is a real number } \}$

Question 15

Find the least common denominator.
-$\frac { x - 6 } { x - 1 } + \frac { x + 2 } { x - 3 }$

Accepted Answer

A) (x - 6)(x + 2) 
B) (x - 1)(x + 2) 
C) (x - 6)(x - 1)(x + 2)(x - 3) 
D) (x - 1)(x - 3) 
A) (x - 6)(x + 2) 
B) (x - 1)(x + 2) 
C) (x - 6)(x - 1)(x + 2)(x - 3) 
D) (x - 1)(x - 3)

Question 16

Add or subtract.
-$\frac { 6 } { x } + \frac { 7 } { x - 8 }$

Accepted Answer

A)  $\frac { 48 x - 13 } { x ( 8 - x ) }$ 
B)  $\frac { 13 x - 48 } { x ( x - 8 ) }$ 
C)  $\frac { 48 x - 13 } { x ( x - 8 ) }$ 
D)  $\frac { 13 x - 48 } { x ( 8 - x ) }$ 
A)  $\frac { 48 x - 13 } { x ( 8 - x ) }$ 
B)  $\frac { 13 x - 48 } { x ( x - 8 ) }$ 
C)  $\frac { 48 x - 13 } { x ( x - 8 ) }$ 
D)  $\frac { 13 x - 48 } { x ( 8 - x ) }$

Question 17

Find the least common denominator.
-$\frac { 5 } { m ^ { 2 } + 6 m } + \frac { 7 } { m ^ { 2 } + 3 m - 18 }$

Accepted Answer

A)  $( m + 9 ) ^ { 2 }$ 
B)  $\mathrm { m } ( \mathrm { m } + 9 ) ^ { 2 }$ 
C)  $m ( m + 6 ) ( m - 3 )$ 
D)  $m ( m + 9 ) ( m - 3 )$ 
A)  $( m + 9 ) ^ { 2 }$ 
B)  $\mathrm { m } ( \mathrm { m } + 9 ) ^ { 2 }$ 
C)  $m ( m + 6 ) ( m - 3 )$ 
D)  $m ( m + 9 ) ( m - 3 )$

Question 18

Multiply.
-$\frac { 3 x ^ { 2 } } { 5 } \cdot \frac { 40 } { x ^ { 3 } }$

Accepted Answer

A)  $\frac { 24 x ^ { 2 } } { x ^ { 3 } }$ 
B)  $\frac { 24 } { x }$ 
C)  $\frac { x } { 24 }$ 
D)  $\frac { 120 x ^ { 2 } } { 5 x ^ { 3 } }$ 
A)  $\frac { 24 x ^ { 2 } } { x ^ { 3 } }$ 
B)  $\frac { 24 } { x }$ 
C)  $\frac { x } { 24 }$ 
D)  $\frac { 120 x ^ { 2 } } { 5 x ^ { 3 } }$

Question 19

Find the least common denominator.
-$\frac { 9 p + 1 } { p ^ { 2 } - 49 } - \frac { p - 1 } { p ^ { 4 } - 7 p ^ { 3 } }$

Accepted Answer

A)  $p ^ { 3 } ( p + 7 ) ( p - 7 ) ^ { 2 }$ 
B)  $p ^ { 3 } ( p - 7 )$ 
C)  $p ^ { 2 } ( p + 7 ) ( p - 7 )$ 
D)  $p ^ { 3 } ( p + 7 ) ( p - 7 )$ 
A)  $p ^ { 3 } ( p + 7 ) ( p - 7 ) ^ { 2 }$ 
B)  $p ^ { 3 } ( p - 7 )$ 
C)  $p ^ { 2 } ( p + 7 ) ( p - 7 )$ 
D)  $p ^ { 3 } ( p + 7 ) ( p - 7 )$

Question 20

Add or subtract.
-$\frac { 8 - x } { x - 6 } - \frac { 2 x - 1 } { 6 - x }$

Accepted Answer

A)  $\frac { x + 9 } { x - 6 }$ 
B)  $- \frac { x + 9 } { x - 6 }$ 
C)  $- \frac { x + 7 } { x - 6 }$ 
D)  $\frac { x + 7 } { x - 6 }$ 
A)  $\frac { x + 9 } { x - 6 }$ 
B)  $- \frac { x + 9 } { x - 6 }$ 
C)  $- \frac { x + 7 } { x - 6 }$ 
D)  $\frac { x + 7 } { x - 6 }$

Simplify the rational expression. - $\frac { y ^ { 2 } + 7 y + 10 } { y ^ { 2 } + 11 y + 30 }$

Divide. - $\frac { x ^ { 2 } - 10 x + 25 } { 7 x - 35 } \div \frac { 2 x - 10 } { 14 }$

Divide. - $\frac { 27 x - 27 } { 5 } \div \frac { 9 x - 9 } { 10 }$

Multiply. - $\frac { 4 x } { 8 x + 4 } \cdot \frac { 6 x + 3 } { 5 }$

Find the least common denominator. - $\frac { 9 } { 4 a + 8 } - \frac { 6 } { a ^ { 2 } + 2 a }$

Add or subtract. - $\frac { 5 y } { x ^ { 2 } - y ^ { 2 } } + \frac { 5 x } { y ^ { 2 } - x ^ { 2 } }$

Simplify the rational expression. - $\frac { 4 x - 14 x ^ { 2 } } { 2 x }$

Add or subtract. - $\frac { 4 x ^ { 2 } + 3 x + 3 } { x ^ { 2 } + 13 x + 40 } - \frac { 3 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } + 13 x + 40 }$

Simplify the rational expression. - $\frac { w ^ { 3 } - 5 w ^ { 2 } - 2 w + 10 } { w ^ { 2 } - 10 w + 25 }$

Multiply. - $\frac { 4 x + y } { - 4 x - 5 y } \cdot \frac { - 4 x ^ { 2 } - 9 x y - 5 y ^ { 2 } } { 4 x ^ { 2 } + 5 x y + y ^ { 2 } }$

Determine the domain of the function. - $r ( x ) = \frac { x ^ { 2 } + x - 20 } { x ^ { 2 } - 14 x + 48 }$

Simplify the rational expression. - $\frac { y ^ { 3 } - 64 } { y - 4 }$

Determine the domain of the function. - $f ( x ) = \frac { 2 x ^ { 2 } - 4 } { 3 x ^ { 2 } + 6 x - 45 }$

Find the least common denominator. - $\frac { x - 6 } { x - 1 } + \frac { x + 2 } { x - 3 }$

Add or subtract. - $\frac { 6 } { x } + \frac { 7 } { x - 8 }$

Find the least common denominator. - $\frac { 5 } { m ^ { 2 } + 6 m } + \frac { 7 } { m ^ { 2 } + 3 m - 18 }$

Multiply. - $\frac { 3 x ^ { 2 } } { 5 } \cdot \frac { 40 } { x ^ { 3 } }$

Find the least common denominator. - $\frac { 9 p + 1 } { p ^ { 2 } - 49 } - \frac { p - 1 } { p ^ { 4 } - 7 p ^ { 3 } }$

Add or subtract. - $\frac { 8 - x } { x - 6 } - \frac { 2 x - 1 } { 6 - x }$

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. - y2+7y+10y2+11y+30\frac { y ^ { 2 } + 7 y + 10 } { y ^ { 2 } + 11 y + 30 }y2+11y+30y2+7y+10​

Divide. - x2−10x+257x−35÷2x−1014\frac { x ^ { 2 } - 10 x + 25 } { 7 x - 35 } \div \frac { 2 x - 10 } { 14 }7x−35x2−10x+25​÷142x−10​

Divide. - 27x−275÷9x−910\frac { 27 x - 27 } { 5 } \div \frac { 9 x - 9 } { 10 }527x−27​÷109x−9​

Solve the problem. -A parallelogram has areaas area x2+9x+8x and height x5x+1\frac { x ^ { 2 } + 9 x + 8 } { x } \text { and height } \frac { x ^ { 5 } } { x + 1 }xx2+9x+8​ and height x+1x5​ . Find the length of its base, by dividing its area by its height.

Multiply. - 4x8x+4⋅6x+35\frac { 4 x } { 8 x + 4 } \cdot \frac { 6 x + 3 } { 5 }8x+44x​⋅56x+3​

Find the least common denominator. - 94a+8−6a2+2a\frac { 9 } { 4 a + 8 } - \frac { 6 } { a ^ { 2 } + 2 a }4a+89​−a2+2a6​

Add or subtract. - 5yx2−y2+5xy2−x2\frac { 5 y } { x ^ { 2 } - y ^ { 2 } } + \frac { 5 x } { y ^ { 2 } - x ^ { 2 } }x2−y25y​+y2−x25x​

Simplify the rational expression. - 4x−14x22x\frac { 4 x - 14 x ^ { 2 } } { 2 x }2x4x−14x2​

Add or subtract. - 4x2+3x+3x2+13x+40−3x2−3x−2x2+13x+40\frac { 4 x ^ { 2 } + 3 x + 3 } { x ^ { 2 } + 13 x + 40 } - \frac { 3 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } + 13 x + 40 }x2+13x+404x2+3x+3​−x2+13x+403x2−3x−2​

Simplify the rational expression. - w3−5w2−2w+10w2−10w+25\frac { w ^ { 3 } - 5 w ^ { 2 } - 2 w + 10 } { w ^ { 2 } - 10 w + 25 }w2−10w+25w3−5w2−2w+10​

Multiply. - 4x+y−4x−5y⋅−4x2−9xy−5y24x2+5xy+y2\frac { 4 x + y } { - 4 x - 5 y } \cdot \frac { - 4 x ^ { 2 } - 9 x y - 5 y ^ { 2 } } { 4 x ^ { 2 } + 5 x y + y ^ { 2 } }−4x−5y4x+y​⋅4x2+5xy+y2−4x2−9xy−5y2​

Determine the domain of the function. - r(x)=x2+x−20x2−14x+48r ( x ) = \frac { x ^ { 2 } + x - 20 } { x ^ { 2 } - 14 x + 48 }r(x)=x2−14x+48x2+x−20​

Simplify the rational expression. - y3−64y−4\frac { y ^ { 3 } - 64 } { y - 4 }y−4y3−64​

Determine the domain of the function. - f(x)=2x2−43x2+6x−45f ( x ) = \frac { 2 x ^ { 2 } - 4 } { 3 x ^ { 2 } + 6 x - 45 }f(x)=3x2+6x−452x2−4​

Find the least common denominator. - x−6x−1+x+2x−3\frac { x - 6 } { x - 1 } + \frac { x + 2 } { x - 3 }x−1x−6​+x−3x+2​

Add or subtract. - 6x+7x−8\frac { 6 } { x } + \frac { 7 } { x - 8 }x6​+x−87​

Find the least common denominator. - 5m2+6m+7m2+3m−18\frac { 5 } { m ^ { 2 } + 6 m } + \frac { 7 } { m ^ { 2 } + 3 m - 18 }m2+6m5​+m2+3m−187​

Multiply. - 3x25⋅40x3\frac { 3 x ^ { 2 } } { 5 } \cdot \frac { 40 } { x ^ { 3 } }53x2​⋅x340​

Find the least common denominator. - 9p+1p2−49−p−1p4−7p3\frac { 9 p + 1 } { p ^ { 2 } - 49 } - \frac { p - 1 } { p ^ { 4 } - 7 p ^ { 3 } }p2−499p+1​−p4−7p3p−1​

Add or subtract. - 8−xx−6−2x−16−x\frac { 8 - x } { x - 6 } - \frac { 2 x - 1 } { 6 - x }x−68−x​−6−x2x−1​