Using the two (2) 7/2x+4) 2 and 3x+2 / (6x+2)(x+5)

Question

ScanSolve AI Tutor · Accepted Answer

Here's the partial fraction decomposition for the expression $\frac{3x+2}{(x+2)(x+5)}$ .

Set up the partial fraction decomposition.
We assume the expression can be written as:
$\frac{3x+2}{(x+2)(x+5)} = \frac{A}{x+2} + \frac{B}{x+5}$
Multiply both sides by $(x+2)(x+5)$ to clear the denominators:
$3x+2 = A(x+5) + B(x+2)$
Solve for $A$ using the substitution method.
To find $A$ , substitute the value of $x$ that makes the term with $B$ zero. This occurs when $x+2=0$ , so $x=-2$ .
Substitute $x=-2$ into the equation $3x+2 = A(x+5) + B(x+2)$ :
$3(-2)+2 = A(-2+5) + B(-2+2)$ $-6+2 = A(3) + B(0)$ $-4 = 3A$ $A = -\frac{4}{3}$
Solve for $B$ using the substitution method.
To find $B$ , substitute the value of $x$ that makes the term with $A$ zero. This occurs when $x+5=0$ , so $x=-5$ .
Substitute $x=-5$ into the equation $3x+2 = A(x+5) + B(x+2)$ :
$3(-5)+2 = A(-5+5) + B(-5+2)$ $-15+2 = A(0) + B(-3)$ $-13 = -3B$ $B = \frac{-13}{-3}$ $B = \frac{13}{3}$
Write the final partial fraction decomposition.
Substitute the values of $A$ and $B$ back into the partial fraction form:
$\frac{3x+2}{(x+2)(x+5)} = \frac{-\frac{4}{3}}{x+2} + \frac{\frac{13}{3}}{x+5}$
This can be written more cleanly as:
$-\frac{4}{3(x+2)} + \frac{13}{3(x+5)}$

Answer