Expand the quotient into partial fractions: (5x − 13) / ((x − 3)(x − 2))

Question

ScanSolve AI Tutor · Accepted Answer

To expand the quotient into partial fractions, we set up the decomposition as follows:

Set up the partial fraction decomposition.
The denominator has two distinct linear factors, $(x-3)$ and $(x-2)$ .
$\frac{5x - 13}{(x - 3)(x - 2)} = \frac{A}{x - 3} + \frac{B}{x - 2}$
Clear the denominators.
Multiply both sides by $(x - 3)(x - 2)$ :
$5x - 13 = A(x - 2) + B(x - 3)$
Solve for A and B.
To find $A$ , let $x = 3$ :
$5(3) - 13 = A(3 - 2) + B(3 - 3)$ $15 - 13 = A(1) + B(0)$ $2 = A$
To find $B$ , let $x = 2$ :
$5(2) - 13 = A(2 - 2) + B(2 - 3)$ $10 - 13 = A(0) + B(-1)$ $-3 = -B$ $B = 3$
Write the final partial fraction decomposition.
Substitute the values of $A$ and $B$ back into the decomposition:
$\frac{5x - 13}{(x - 3)(x - 2)} = \frac{2}{x - 3} + \frac{3}{x - 2}$
The expanded quotient is:
$\frac{2}{x - 3} + \frac{3}{x - 2}$

Expand the quotient into partial fractions: (5x - 13) / ((x - 3)(x - 2))