# How do you express  (3x+5)/((3x+2)(2x+1) ) in partial fractions?

Jul 22, 2018

Here ,
$7 \left(3 x + 2\right) - 9 \left(2 x + 1\right) = 21 x + 14 - 18 x - 9 = 3 x + 5$

$\frac{3 x + 5}{\left(3 x + 2\right) \left(2 x + 1\right)} = \frac{7 \left(3 x + 2\right) - 9 \left(2 x + 1\right)}{\left(3 x + 2\right) \left(2 x + 1\right)}$

$\frac{3 x + 5}{\left(3 x + 2\right) \left(2 x + 1\right)} = \frac{7}{2 x + 1} - \frac{9}{3 x + 2}$

#### Explanation:

Let ,

color(red)((3x+5)/((3x+2)(2x+1))=A/(3x+2)+B/(2x+1)to(X)

$\therefore 3 x + 5 = A \left(2 x + 1\right) + B \left(3 x + 2\right)$

$\implies 3 x + 5 = 2 x A + A + 3 x B + 2 B$

$\implies 3 x + 5 = 2 x A + 3 x B + A + 2 B$

$\implies 3 x + 5 = x \left(2 A + 3 B\right) + \left(A + 2 B\right)$

Comparing coefficient of $x \mathmr{and}$ constant term :

color(blue)(2A+3B=3 to(1) and A+2B=5 to (2)

From $\left(2\right) ,$ we get color(blue)( A=5-2B to(3)

Subst. $A = 5 - 2 B$ into $\left(1\right)$

$2 \left(5 - 2 B\right) + 3 B = 3$

$\therefore 10 - 4 B + 3 B = 3$

$\therefore - 4 B + 3 B = 3 - 10$

$\therefore - B = - 7$

=>color(violet)(B=7

Subst. $B = 7$ into $\left(3\right)$

$A = 5 - 2 \left(7\right) = 5 - 14$

:.color(violet)(A=-9

Subst. $A = - 9 \mathmr{and} B = 7$ into $\left(X\right)$

color(red)((3x+5)/((3x+2)(2x+1))=(-9)/(3x+2)+7/(2x+1)

OR

$\frac{3 x + 5}{\left(3 x + 2\right) \left(2 x + 1\right)} = \frac{7}{2 x + 1} - \frac{9}{3 x + 2}$

Jul 22, 2018

The answer is $= - \frac{9}{3 x + 2} + \frac{7}{2 x + 1}$

#### Explanation:

Perform the decomposition into partial fractions

$\frac{3 x + 5}{\left(3 x + 2\right) \left(2 x + 1\right)} = \frac{A}{3 x + 2} + \frac{B}{2 x + 1}$

$= \frac{A \left(2 x + 1\right) + B \left(3 x + 2\right)}{\left(3 x + 2\right) \left(2 x + 1\right)}$

Compare the numerators

$3 x + 5 = A \left(2 x + 1\right) + B \left(3 x + 2\right)$

Let $x = - \frac{2}{3}$

Then,

$3 = - \frac{1}{3} A$

$\implies$, $A = - 9$

Let $x = - \frac{1}{2}$

Then,

$\frac{7}{2} = \frac{1}{2} B$

$\implies$, $B = 7$

Finally,

$\frac{3 x + 5}{\left(3 x + 2\right) \left(2 x + 1\right)} = - \frac{9}{3 x + 2} + \frac{7}{2 x + 1}$