Topic 1.4 - Partial Fractions

TOPIC 1.4: Partial Fractions A sum of two algebraic fractions can be expresses as a single fraction: 3( x  1)  x(3 x  2) 3 x  3  3 x 2  2 x 3 x 3  5 x  3x 2    = 3x  2 x  1 (3 x  2)( x  1) (3 x  2)( x  1) (3 x  2)( x  1) Now we are interested in the reverse process called Partial fractions by which a single algebraic fraction is expressed as a sum of two or more simpler fractions. Patterns: Pattern 1

Partial fractions with simple denominators x 1 A B C =   x (2 x  1)( x  2) x 2x  1 x  2

Pattern 2

Partial fractions when the denominator has a repeated factor x5 A B C =   2 x  1 x  1 ( x  1) 2 ( x  1)( x  1)

Pattern 3

Partial fractions when the denominator includes a quadratic factor A Bx  C 2x  3 =  2 2 3x  2 x  1 (3 x  2)( x  1)

Pattern 4

Partial fractions of improper fractions B C x2  2 = A  x  1 2x  1 ( x  1)(2 x  1)

Example 1: Partial fractions with simple denominators 1. Express the following fraction in partial fractions. 5 5 x 2  12 x  5 (a) 2 (b) 2 x  x6 ( x  1)( x  2)

Example 2: Partial fractions when denominator includes a repeated factor x 2  13 x 2  11 (a) (b) ( x  1) 2 ( x  2) ( x  2) 2 (3x  1)

Example 3: Partial fractions when denominator includes a quadratic factor x2 5x (a) (b) 2 (2 x  1)( x  1) ( x  2)( x 2  1) Example 4: Partial fractions of improper fractions 3x 2 x3  x2 1 (a) 2 (b) x x2 x( x 2  1)

(c)

x 3  2 x 2  3x  6 x 2 (2  x)