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Chapter 11: Q. 47 (page 765)

Use the division algorithm to rewrite the improper fraction as the sum of a quotient and proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition.

x3+x2-3x2+3x-4

Short Answer

Expert verified

The partial fraction decomposition of the given rational expression is given below,

515x+4+-15x-1

On expression the improper fraction as the sum of a quotient and the partial fraction decomposition,

x-2+10x-11x2+3x-4

Step by step solution

01

Step 1. Given information.

Consider the given question,

x3+x2-3x2+3x-4

02

Step 2. Use the division algorithm.

Using the division algorithm,

Therefore, it can be said,

x3+x2-3x2+3x-4=x-2+10x-11x2+3x-4.

03

Step 3. Factor the denominator.

The denominator can be factorized as x2+3x-4=x+4x-1.

The denominator has a repeated non-reducible quadratic factors. Then we decompose the rational expression as follows,

localid="1647597527675" 10x-11x2+3x-4=10x-11x+4x-110x-11x2+3x-4=Ax+4+Bx-1......(i)

Then,

10x-11=Ax-1+Bx+410x-11=Ax-A+Bx+4B10x-11=A+Bx+-A+4B

Therefore, it can be written,

localid="1647597550459" A+B=10......(ii)-A+4B=-11......(iii)

04

Step 4. Add equations (ii) and (iii).

On adding equations (ii) and (iii),

A+B+-A+4B=10-115B=-1B=-15

Substitute the value of Bin equation (ii),

A-15=10A=515

Substitute the values in equation (i),

10x-11x2+3x-4=515x+4+-15x-1

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