Question

Write out the form of the partial fraction decomposition of the function (See Example). Do not determine the numerical values of the coefficients. (If the partial fraction decomposition does not exist, enter DNE.) (a) x4 + 7 x5 + 5x3 A x​+ B x2​+ C x3​+ Dx+E x2+5​ (b) 2 (x2 − 9)2

Answer 1

Answer:

a.$\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x^2+5}$

b.$\frac{A}{x+3}+\frac{B}{x-3}+\frac{C}{(x-3)^2}+\frac{D}{(x+3)^2}$

Explanation:

a.We are given that

$\frac{x^4+7}{x^5+5x^3}$

$\frac{x^4+7}{x^5+5x^3}=\frac{x^4+7}{x^3(x^2+5)}$

Using partial fraction decomposition of the given function

$\frac{x^4+7}{x^3(x^2+5)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x^2+5}$

Using the formula

$\frac{1}{x^3(x^2+a)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x^2+a}$

b.$\frac{2}{(x^2-9)^2}$

$\frac{2}{(x^2-3^2)^2}=\frac{2}{(x+3)^2(x-3)^2}$

Using property $a^2-b^2=(a+b)(a-b)$

$\frac{2}{(x+3)^2(x-3)^2}=\frac{A}{x+3}+\frac{B}{x-3}+\frac{C}{(x-3)^2}+\frac{D}{(x+3)^2}$

Using the property

$\frac{1}{x^2}=\frac{A}{x}+\frac{B}{x^2}$