Alg5.1.5

Write as sum of partial fractions $\frac{5x+2}{(1+3x)(1+2x)}\,$

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

Multiply all terms on both sides by the common denominator $(1+3x)(1+2x)\,$ .

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

So if $x=\frac{-1}{2}\,$

then $5\left[\frac{-1}{2}\right]+2=B(1+3\left[\frac{-1}{2}\right])\,$ .

$\frac{-1}{2}=\frac{-B}{2}\,$ .

Therefore $B=1\,$ .

Similarly if $x=\frac{-1}{3}\,$

then $5\frac{-1}{3}+2=A(1+2\frac{-1}{3}\,$ .

then $\frac{1}{3}=\frac{A}{3}\,$ .

Therefore $A=1\,$ .

So $\frac{5x+2}{(1+3x)(1+2x)} = \frac{1}{1+3x} + \frac{1}{1+2x}\,$