How do you express $x+22 div x^2+2x-8$ in partial fractions?

Question

1830 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-09T09:37:32

Partial fractions of $(x+2)/((x-2)(x+4))$ are $4/(x-2)-3/(x+4)$

To express $(x+22)/(x^2+2x-8)$ in partial fractions, first let us factorize the denominator $(x^2+2x-8)$ . This can be done as follows:

$x^2+2x-8=x^2+4x-2x-8)=x(x+4)-2(x+4)=(x-2)(x+4)$

Hence $(x+22)/(x^2+2x-8)$ can be written as $(x+22)/((x-2)(x+4))$ .

Let the partial fractions be $(x+22)/((x-2)(x+4))=A/(x-2)+B/(x+4)$

Simplifying RHS, we get $(A(x+4)+B(x-2))/((x-2)(x+4))$ or

$((A+B)x+(4A-2B))/((x-2)(x+4))$ and as this is equivalent to $(x+2)/((x-2)(x+4))$ , we have

$A+B=1$ and $4A-2B=22$ . From first we get, $B=1-A$ and putting this in second we get $4A-2(1-A)=22$ or $6A=24$ or $A=4$ and hence $B=1-4=-3$

Hence partial fractions of $(x+2)/((x-2)(x+4))$ are $4/(x-2)-3/(x+4)$

How do you express x+22 div x^2+2x-8x+22 div x^2+2x-8 in partial fractions?