Question 1

Write the partial fraction decomposition of the rational expression.Check your result algebraically.​ $\frac { x + 1 } { x ^ { 2 } - 2 x - 8 }$ ​

Accepted Answer

A)  $\frac { 1 } { 6 } \left( \frac { 2 } { x - 4 } - \frac { 4 } { x + 2 } \right)$ 
B)  $$\frac { 1 } { 2 } \left( \frac { 5 } { x - 4 } - \frac { 1 } { x + 2 } \right)$$ 
C)  $\frac { 1 } { 6 } \left( \frac { 5 } { x - 4 } - \frac { 1 } { x + 2 } \right)$ 
D)  $\frac { 1 } { 4 } \left( \frac { 5 } { x - 4 } - \frac { 1 } { x + 2 } \right)$ 
E)  $\frac { 1 } { 6 } \left( \frac { 5 } { x - 4 } + \frac { 1 } { x + 2 } \right)$ 
A)  $\frac { 1 } { 6 } \left( \frac { 2 } { x - 4 } - \frac { 4 } { x + 2 } \right)$ 
B)  $$\frac { 1 } { 2 } \left( \frac { 5 } { x - 4 } - \frac { 1 } { x + 2 } \right)$$ 
C)  $\frac { 1 } { 6 } \left( \frac { 5 } { x - 4 } - \frac { 1 } { x + 2 } \right)$ 
D)  $\frac { 1 } { 4 } \left( \frac { 5 } { x - 4 } - \frac { 1 } { x + 2 } \right)$ 
E)  $\frac { 1 } { 6 } \left( \frac { 5 } { x - 4 } + \frac { 1 } { x + 2 } \right)$

Question 2

Write the partial fraction decomposition of the rational expression.​ $\frac { 7 x ^ { 2 } - 3 x - 6 } { x ^ { 3 } - x }$ ​

Accepted Answer

A)  $\frac { 6 } { x } + \frac { 2 } { x - 1 } - \frac { 1 } { x - 1 }$ 
B)  $\frac { 6 } { x } + \frac { 2 } { x + 1 } + \frac { 1 } { x - 1 }$ 
C)  $\frac { 6 } { x } - \frac { 2 } { x + 1 } - \frac { 1 } { x - 1 }$ 
D)  $\frac { 6 } { x } + \frac { 2 } { x + 1 } - \frac { 1 } { x + 1 }$ 
E)  $\frac { 6 } { x } + \frac { 2 } { x + 1 } - \frac { 1 } { x - 1 }$ 
A)  $\frac { 6 } { x } + \frac { 2 } { x - 1 } - \frac { 1 } { x - 1 }$ 
B)  $\frac { 6 } { x } + \frac { 2 } { x + 1 } + \frac { 1 } { x - 1 }$ 
C)  $\frac { 6 } { x } - \frac { 2 } { x + 1 } - \frac { 1 } { x - 1 }$ 
D)  $\frac { 6 } { x } + \frac { 2 } { x + 1 } - \frac { 1 } { x + 1 }$ 
E)  $\frac { 6 } { x } + \frac { 2 } { x + 1 } - \frac { 1 } { x - 1 }$

Question 3

Write the form of the partial fraction decomposition of the rational expression.​ $$\frac { 5 x - 1 } { x \left( x ^ { 2 } + 5 \right) }$$ ​

Accepted Answer

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

Question 4

Write the partial fraction decomposition of the rational expression.​ $\frac { 1 } { a ^ { 2 } - x ^ { 2 } }$ ​

Accepted Answer

A)  $\frac { 1 } { 2 a } \left( \frac { 1 } { a + x } - \frac { 1 } { a - x } \right)$ 
B)  $\frac { a } { a + x } + \frac { a } { a - x }$ 
C)  $\frac { 1 } { 2 a } \left( \frac { 1 } { a + x } + \frac { 1 } { a - x } \right)$ 
D)  $\frac { 1 } { 2 a } \left( \frac { 1 } { a - x } + \frac { 1 } { ( a - x ) ^ { 2 } } \right)$ 
E)  $\frac { 1 } { a } \left( \frac { 1 } { a + x } + \frac { 1 } { a - x } \right)$ 
A)  $\frac { 1 } { 2 a } \left( \frac { 1 } { a + x } - \frac { 1 } { a - x } \right)$ 
B)  $\frac { a } { a + x } + \frac { a } { a - x }$ 
C)  $\frac { 1 } { 2 a } \left( \frac { 1 } { a + x } + \frac { 1 } { a - x } \right)$ 
D)  $\frac { 1 } { 2 a } \left( \frac { 1 } { a - x } + \frac { 1 } { ( a - x ) ^ { 2 } } \right)$ 
E)  $\frac { 1 } { a } \left( \frac { 1 } { a + x } + \frac { 1 } { a - x } \right)$

Question 5

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants.​ $$\frac { x ^ { 2 } - 9 x + 20 } { 7 x ^ { 3 } + 19 x ^ { 2 } }$$ ​

Accepted Answer

A)  $\frac { A } { x } + \frac { B } { 7 x ^ { 3 } + 19 x ^ { 2 } }$ 
B)  $\frac { A } { x } + \frac { B } { 7 x ^ { 2 } + 19 }$ 
C)  $\frac { A } { x } + \frac { B } { x ^ { 2 } - 9 x + 20 } + \frac { C } { 7 x ^ { 2 } + 19 }$ 
D)  $\frac { A } { 9 x ^ { 3 } } - \frac { B } { 19 x ^ { 2 } } - \frac { C } { 7 x + 19 }$ 
E) ​ $\frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C } { 7 x + 19 }$ 
A)  $\frac { A } { x } + \frac { B } { 7 x ^ { 3 } + 19 x ^ { 2 } }$ 
B)  $\frac { A } { x } + \frac { B } { 7 x ^ { 2 } + 19 }$ 
C)  $\frac { A } { x } + \frac { B } { x ^ { 2 } - 9 x + 20 } + \frac { C } { 7 x ^ { 2 } + 19 }$ 
D)  $\frac { A } { 9 x ^ { 3 } } - \frac { B } { 19 x ^ { 2 } } - \frac { C } { 7 x + 19 }$ 
E) ​ $\frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C } { 7 x + 19 }$

Question 6

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants.​ $$\frac { 5 x - 4 } { x ^ { 3 } + 5 x }$$ ​

Accepted Answer

A)  $\frac { A } { 5 x - 4 } + \frac { B x + C } { x ^ { 3 } + 5 x }$ 
B)  $\frac { A } { x } - \frac { B x + C } { x ^ { 2 } + 5 } + \frac { D } { x ^ { 3 } + 5 }$ 
C)  $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 5 }$ 
D)  $\frac { A } { x } + \frac { B x - C } { x ^ { 2 } + 5 } + \frac { D } { 5 x - 4 }$ 
E)  $\frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { B x + C } { x + 5 }$ 
A)  $\frac { A } { 5 x - 4 } + \frac { B x + C } { x ^ { 3 } + 5 x }$ 
B)  $\frac { A } { x } - \frac { B x + C } { x ^ { 2 } + 5 } + \frac { D } { x ^ { 3 } + 5 }$ 
C)  $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 5 }$ 
D)  $\frac { A } { x } + \frac { B x - C } { x ^ { 2 } + 5 } + \frac { D } { 5 x - 4 }$ 
E)  $\frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { B x + C } { x + 5 }$

Question 7

Write the partial fraction decomposition of the rational expression.Check your result algebraically.​ $$\frac { 5 } { x ^ { 2 } - 5 x }$$ ​

Accepted Answer

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

Question 8

Write the partial fraction decomposition of the rational expression. $\frac { - 3 x ^ { 2 } - 3 x + 27 } { x ^ { 2 } ( x + 3 ) }$

Accepted Answer

A)  $\frac { 1 } { x + 3 } + \frac { 9 } { x ^ { 2 } }$ 
B)  $\frac { 1 } { x + 3 } - \frac { 4 } { x } + \frac { 2 } { x ^ { 2 } }$ 
C)  $\frac { 1 } { x + 3 } - \frac { 4 } { x } + \frac { 9 } { x ^ { 2 } }$ 
D)  $\frac { 1 } { x + 3 } + \frac { 4 } { x } - \frac { 9 } { x ^ { 2 } }$ 
E)  $\frac { - 3 } { x + 3 } - \frac { 3 } { x ( x + 3 ) } + \frac { 27 } { x ^ { 2 } ( x + 3 ) }$ 
A)  $\frac { 1 } { x + 3 } + \frac { 9 } { x ^ { 2 } }$ 
B)  $\frac { 1 } { x + 3 } - \frac { 4 } { x } + \frac { 2 } { x ^ { 2 } }$ 
C)  $\frac { 1 } { x + 3 } - \frac { 4 } { x } + \frac { 9 } { x ^ { 2 } }$ 
D)  $\frac { 1 } { x + 3 } + \frac { 4 } { x } - \frac { 9 } { x ^ { 2 } }$ 
E)  $\frac { - 3 } { x + 3 } - \frac { 3 } { x ( x + 3 ) } + \frac { 27 } { x ^ { 2 } ( x + 3 ) }$

Exam 44: Partial Fractions

Write the partial fraction decomposition of the rational expression.Check your result algebraically.​ x+1x2−2x−8\frac { x + 1 } { x ^ { 2 } - 2 x - 8 }x2−2x−8x+1​ ​

Write the partial fraction decomposition of the rational expression.​ 7x2−3x−6x3−x\frac { 7 x ^ { 2 } - 3 x - 6 } { x ^ { 3 } - x }x3−x7x2−3x−6​ ​

Write the form of the partial fraction decomposition of the rational expression.​ 5x−1x(x2+5)\frac { 5 x - 1 } { x \left( x ^ { 2 } + 5 \right) }x(x2+5)5x−1​ ​

Write the partial fraction decomposition of the rational expression.​ 1a2−x2\frac { 1 } { a ^ { 2 } - x ^ { 2 } }a2−x21​ ​

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants.​ x2−9x+207x3+19x2\frac { x ^ { 2 } - 9 x + 20 } { 7 x ^ { 3 } + 19 x ^ { 2 } }7x3+19x2x2−9x+20​ ​

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants.​ 5x−4x3+5x\frac { 5 x - 4 } { x ^ { 3 } + 5 x }x3+5x5x−4​ ​

Write the partial fraction decomposition of the rational expression.Check your result algebraically.​ 5x2−5x\frac { 5 } { x ^ { 2 } - 5 x }x2−5x5​ ​

Write the partial fraction decomposition of the rational expression. −3x2−3x+27x2(x+3)\frac { - 3 x ^ { 2 } - 3 x + 27 } { x ^ { 2 } ( x + 3 ) }x2(x+3)−3x2−3x+27​

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Write the partial fraction decomposition of the rational expression.Check your result algebraically. $\frac { x + 1 } { x ^ { 2 } - 2 x - 8 }$

Write the partial fraction decomposition of the rational expression. $\frac { 7 x ^ { 2 } - 3 x - 6 } { x ^ { 3 } - x }$

Write the form of the partial fraction decomposition of the rational expression. $\frac { 5 x - 1 } { x \left( x ^ { 2 } + 5 \right) }$

Write the partial fraction decomposition of the rational expression. $\frac { 1 } { a ^ { 2 } - x ^ { 2 } }$

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants. $\frac { x ^ { 2 } - 9 x + 20 } { 7 x ^ { 3 } + 19 x ^ { 2 } }$

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants. $\frac { 5 x - 4 } { x ^ { 3 } + 5 x }$

Write the partial fraction decomposition of the rational expression.Check your result algebraically. $\frac { 5 } { x ^ { 2 } - 5 x }$

Write the partial fraction decomposition of the rational expression. $\frac { - 3 x ^ { 2 } - 3 x + 27 } { x ^ { 2 } ( x + 3 ) }$