Question

Find the Partical Fraction Decomposition and show your work.​

Answer 1

Expand the rational expression as

`\frac ax+(bx+c)/(x^2+1)+\frac d{x-5}+\frac e{x+5}`

Then

`2x^4+3x^3-8x^2-9x-10=a(x^2+1)(x^2-5)+(bx+c)x(x^2-5)+dx(x^2+1)(x+5)+ex(x^2+1)(x-5)`

Expand the right side, then set the coefficients of each term on both sides equal to each other and solve for the coefficients.

`ax^4-4ax^2-5a+bx^4+cx^3-5bx^2-5cx+dx^4+5dx^3+dx^2+5dx+ex^4-5ex^3+ex^2-5ex`

`=(a+b+d+e)x^4+(c+5d-5e)x^3+(-4a-5b+d+e)x^2+(-5c+5d-5e)x-5a`

`\implies\begin{cases}a+b+d+e=2\\c+5d-5e=3\\-4a-5b+d+e=-8\\-5c+5d-5e=-9\\-5a=-10\end{cases}\implies\begin{cases}a=2\\b=0\\c=2\\d=\frac1{10},e=-\frac1{10}\end{cases}`

So the expansion is

`\frac2x+\frac2{x^2+1}+\frac1{10x-50}-\frac1{10x+50}`