Question

determine the partial fraction decom

1. \\(\frac{17x - 53}{x^2 - 2x - 15}\\) solution

Explanation:

Step1: Factor the denominator

Factor $x^2 - 2x - 15$ into linear terms:
$x^2 - 2x - 15 = (x-5)(x+3)$

Step2: Set up partial fractions

Express the rational function as sum of two fractions:
$\frac{17x - 53}{(x-5)(x+3)} = \frac{A}{x-5} + \frac{B}{x+3}$

Step3: Eliminate denominators

Multiply both sides by $(x-5)(x+3)$:
$17x - 53 = A(x+3) + B(x-5)$

Step4: Solve for A and B

First, substitute $x=5$:
$17(5) - 53 = A(5+3) + B(5-5)$
$85 - 53 = 8A$
$32 = 8A \implies A=4$

Then substitute $x=-3$:
$17(-3) - 53 = A(-3+3) + B(-3-5)$
$-51 - 53 = -8B$
$-104 = -8B \implies B=13$

Step5: Substitute A and B back

Replace A and B in the partial fraction form.

Answer:

$\frac{4}{x-5} + \frac{13}{x+3}$