Question

express the following in terms of 3 parital fractions:
(2-5x^2) / (2-x)(1+x^2)

Answer 1

otherwise written as
\[\frac{ 2-5x^2 }{ (2-x)(1+x)^2 }\]

Answer 2

so far, i have for the numerator:
2-5x^2 = A(1+x)(1+x)^2 + B(2-x)(1+x)^2 + C(2-x)(1+x)

Answer 3

Your placement of the exponent is confusing. If \(1+x\) is being squared, you have
\[\frac{2-5x^2}{(2-x)(1+x)^2}=\frac{A}{2-x}+\frac{B}{1+x}+\frac{C}{(1+x)^2}\]
If you meant to have \(1+x^2\), then you have
\[\frac{2-5x^2}{(2-x)(1+x^2)}=\frac{A}{2-x}+\frac{Bx+C}{1+x^2}\]

Answer 4

In the first case, distributing all the factors gives
\[\begin{align*}
2-5x^2&=A(1+x)^2+B(2-x)(1+x)+C(2-x)\\
&=(A-B)x^2+(2A+B-C)x+A+2B+2C
\end{align*}\]
In the second,
\[\begin{align*}
2-5x^2&=A(1+x^2)+(Bx+C)(2-x)\\
&=(A-B)x^2+(2B-C)x+A+2C
\end{align*}\]