Write down the form of partial fraction decomposition for th

Write down the form of partial fraction decomposition for the following rational functions but do not work out the coefficients. Note that you should factorise the denominators as far as possible and check if you end up with a quadratic factor that does not factorise into two linear factors. x^7 + 1/(x^4 - 2x^2 + 1)^2 x^7 + 1/(x^4 + 2x^2 + 1)^2

Solution

a. denominator (x² - 1)² (x²- 1)² = ( x+1)⁴ (x -1)⁴

^{partial fraction is} = A / (x+1) + B / (x+1)²+ C /(x+1)³ + D / (x+1)⁴ + e /(x-1) +f (*x-1)²+ g/(x-1)³+ h/(x-1)⁴

^b denominator is (x²+1)⁴ =( Ax+B) / x²+1 + ( Cx +D) / (x²+1)² + (ex+f)/ (x²+1)³ +(gx +h)/(x²+1)⁴