# PARTIAL FRACTION WORKSHEET FOR GRADE 11

Partial Fraction Worksheet for Grade 11 :

Here we are going to see some practice questions on partial fractions.

A rational expression f(x)/g(x) is called a proper fraction if the degree of f(x) is less than degree of g(x), where g(x) can be factored into linear factors and quadratic factors without real zeros. Now f(x) g(x) can be expressed in simpler terms, namely, as a sum of expressions of the form

How to resolve the given linear fraction as partial fraction ?

First we have to factorize the denominator into prime factors.There are three types in partial fraction.

## Linear factors, no factor is repeated

When the factors of the denominator of the given fraction are all linear factors none of which is repeated. We write the partial fraction as follows.

(x +3)/(x + 1) (x - 2)

here the denominator is in the form linear factors  and no factor is repeated. So we can write the partial-fraction as

(x +3) / (x + 1) (x - 2) = [A/(x + 1)] + [B/(x - 2)]

where A and B are constants.

## Linear factors, Some of the factors are repeated

When the factors of the denominator of the given fraction are all linear factors none of which is repeated. We write the partial fraction as follows.

If a linear factor (a x + b) occurs n times as a factor of the denominator of the given fraction, then we can write the partial-fraction as

(x + 3)/(x - 2)³

= [A/(x - 2)] + [B/(x - 2)²] + [C/(x - 2)³]

where A, B and C are constants.

## Quadratic equation in the denominator

If a quadratic equation a x² + b x + c which is not favorable into linear factors occurs only once as factor of the denominator of the given fraction, then we can write the partial fraction as

Based on the above types, you may try the problems given below.

## Partial Fraction Worksheet for Grade 11 - Problems

Resolve the following rational expressions into partial fractions.

(1)  1/(x2-a2)               Solution

(2)  (3x + 1)/(x - 2) (x  + 1)        Solution

(3)  x/(x2 + 1)(x - 1)(x + 2)      Solution

(4)  x/(x-1)3           Solution

(5)  1/(x4 - 1)       Solution

(6)  (x - 1)2 / (x3 + x)     Solution

(7)  (x2 + x + 1)/(x2 - 5x + 6)     Solution

(8)  (x3 + 2x + 1)/(x2 + 5x + 6)   Solution

(9)  (x + 12) / (x + 1)2 (x - 2)    Solution

 (10) Solution (11) (12)  (7 + x) / (1 + x)(1 + x2)      Solution After having gone through the stuff given above, we hope that the students would have understood, how to decompose a rational expression into partial fractions.

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