Question

Write the partial fraction decomposition of the given rational expression. startfraction 5 x plus 2 over x superscript 4 baseline minus 4 x squared endfraction

Answer 1

Answer:

Step-by-step explanation:

Given is a rational algebraic expression in x as

$\frac{5x+2}{x^4-4x^2}$

First let us factorize the denominator.

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

Let the given term be

$\frac{A}{x} +\frac{B}{x^2} +\frac{C}{x-2} +\frac{c}{x+2} \\\\=\frac{Ax(x^2-4)+B(x^2-4)+cx^2(x+2)+Dx^2(x-2)}{x^4-4x^2}$

Equate the numerator to get

$Ax(x^2-4)+B(x^2-4)+cx^2(x+2)+Dx^2(x-2)$=$5x+2$

Substitute x =-2

$4D (-4) = -8\\D=\frac{1}{2}$

Now substitute x =2

We get

$16C=12\\C=\frac{3}{4}$

Put x=0

$-4B=2\\B=\frac{-1}{2}$

Next equate X cube terms to 0 on right side

$A+C+D=0\\A=\frac{-5}{4}$

Hence partial fraction is

$\frac{-1}{2x^2} -\frac{5}{4x} +\frac{1}{2(x+2)} +\frac{3}{4(x-2)}$

Answer 2

The answer is not that hard it is 160