   Chapter 8, Problem 12PS

Chapter
Section
Textbook Problem

Partial Fraction Decomposition Use the result of Exercise 11 to find the partial fraction decomposition of x 3 − 3 x 2 + 1 x 4 − 13 x 2 + 12 x .

To determine

To calculate: The partial fraction decomposition of x33x2+1x413x2+12x.

Explanation

Given:

The function Pk=N(ck)D(ck) where N is a polynomial degree less than n and k=1,2,3,...,n, which is partial fraction decomposition of N(x)D(x).

Here, the given function is x33x2+1x413x2+12x.

Formula used:

If N is a polynomial of degree less than n, then

N(x)D(x)=P1xc1+P2xc2+...+Pnxcn

Calculation:

The value of n is 4.

So, according to the formula,

x33x2+1x413x2+12x=P1x+P2x1+P3x+4+P4x3 …… (1)

This is obtained by factorising x413x2+12x which gives x,x1,x+4,x3

Thus, on comparison with the formula, c1=0,c2=1,c3=4,c4=3

N(x)=x33x2+1

The derivative of D(x) is calculated by using the formula ddx(xn)=nxn1

D(x)=ddx(x413x2+12x)=4x326x+12

Next, put Pk=N(ck)D

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