Aims:• To be able to solve partial fractions with repeated factors• To be able to spot and cancel down in improper fraction before splitting it into it’s partial fractions

Algebra - Partial Fractions Lesson 4

Plenary on white boards? http://integralmaths.org/resources/file.php/41/js/AlgPartialFrac.tst

Denominators with a repeated linear factor

In this case, the partial fractions will be of the form:

This is an example of a fraction whose denominator contains a repeated linear factor.

2

2 2

6 2 + +( + 4)( 3) + 4 3 ( 3)

x x A B Cx x x x x

2

2

6 2( + 4)( 3)

x xx x

We can now find A, B and C using a combination of substitution and equating the coefficients.

To find B we can switch to the method of comparing coefficients.

Substitute x = into :1

Substitute x = into :1

Denominators with a repeated linear factorMultiply through by (x + 4)(x – 3)2

1

Therefore

2

13 +12(4 )x

x x

Let 2 2

13 +12 + +(4 ) 4x A B C

x x x x x

Equate the coefficients of x2 in :1

Denominators with a repeated linear factor

Multiply through by x2(4 – x):

Substitute x = into :1

Substitute x = into :1

1

Denominators with a repeated linear factor

We can find A by comparing the coefficients of x2.

Therefore

Denominators with a repeated linear factor

On w/b express as P.F.Do exercise 1 on the worksheet15 mins

221 xxx

222

21

11

xxx

Improper fractions

Remember, an algebraic fraction is called an improper fraction if the degree of the polynomial is equal to, or greater than, the degree of the denominator.

To express an improper fraction in partial fractions we start by expressing it in the algebraic equivalent of mixed number form.

2

2

2 3 +132 15

x xx x

Using long division:

Improper fractions

Therefore2

2

2 3 +13 5 62 +2 15 ( + 3) ( 5)

x xx x x x

On w/b 1. Do exercise 2 on worksheet.24 3

( 3)( 2)x

x x

39 194( 3) ( 2)x x