004 Sample Final A, Problem 10

Decompose into separate partial fractions.      ${\frac {6x^{2}+27x+31}{(x+3)^{2}(x-1)}}$

Foundations
1) What is the form of the partial fraction decomposition of ${\frac {3x-37}{(x+1)(x-4)}}$?
2) What is the form of the partial fraction decomposition of ${\frac {4x^{2}}{(x-1){(x-2)}^{2}}}$?
Answer:
1) ${\frac {A}{x+1}}+{\frac {B}{x-4}}$
2)${\frac {A}{x-1}}+{\frac {B}{x-2}}+{\frac {C}{{(x-2)}^{2}}}$

Solution:

Step 1:
We set ${\frac {6x^{2}+27x+31}{(x+3)^{2}(x-1)}}={\frac {A}{x-1}}+{\frac {B}{x+3}}+{\frac {C}{{(x+3)}^{2}}}$.

Step 2:
Multiplying both sides of the equation by $(x+3)^{2}(x-1)$, we get
$6x^{2}+27x+31=A(x+3)^{2}+B(x+3)(x-1)+C(x-1)$.

Step 3:

Step 4:

Final Answer:

Return to Sample Exam