How do you express $\frac{x + 2}{(2 x + 7) (x + 1)}$ $(x + 2)/((2x+7)(x+1))$ in partial fractions?

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How do you express $\frac{x + 2}{(2 x + 7) (x + 1)}$ $(x + 2)/((2x+7)(x+1))$ in partial fractions?

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Answer 1 · 2016-11-12T02:41:24

$\frac{A}{2 x + 7} + \frac{B}{x + 1} = \frac{x + 2}{(2 x + 7) (x + 1)}$ $A/(2x+ 7) + B/(x + 1) = (x+ 2)/((2x + 7)(x + 1))$

Put on a common denominator.

$\frac{A (x + 1)}{(2 x + 7) (x + 1)} + \frac{B (2 x + 7)}{(2 x + 7) (x + 1)} = \frac{x + 2}{(2 x + 7) (x + 1)}$ $(A(x + 1))/((2x+ 7)(x + 1)) + (B(2x + 7))/((2x + 7)(x + 1)) = (x+2)/((2x+ 7)(x + 1))$

$A x + A + 2 B x + 7 B = x + 2$ $Ax + A + 2Bx + 7B = x+ 2$

$(A + 2 B) x + (A + 7 B) = x + 2$ $(A + 2B)x + (A + 7B) = x + 2$

So, $A + 2 B = 1$ $A + 2B = 1$ and $A + 7 B = 2$ $A + 7B = 2$ .

Solve.

$A = 1 - 2 B \to 1 - 2 B + 7 B = 2$ $A = 1 - 2B -> 1 - 2B + 7B = 2$

$5 B = 1$ $5B = 1$

$B = \frac{1}{5}$ $B = 1/5$

$:.A = 1 - 2(1/5) = 3/5$

Hence, the partial fraction decomposition is $3/(5(2x + 7)) + 1/(5(x+ 1)) = (x + 2)/((2x + 7)(x + 1))$ .

Hopefully this helps!

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How do you express $\frac{x + 2}{(2 x + 7) (x + 1)}$ $(x + 2)/((2x+7)(x+1))$ in partial fractions?

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