Understanding Partial Fraction Decomposition

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The partial fraction decomposition method is a powerful technique used to simplify rational functions by breaking them into simpler fractions. It involves reducing the degree of either the numerator or the denominator. Learn about proper and improper fractions, simple and repeated factors, and how to express expressions in partial fractions.

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Partial Fraction

Partial Fractions The partial fraction decomposition or partial fraction expansion is used to reduce the degree of either the numerator or the denominator of a rational function/algebraic fraction. Proper and Improper Fraction Revisited The fraction is proper, if degree of the denominator > degree of the numerator 3 2+ x x Eg. 4 5 The fraction is improper, if degree of the denominator degree of the numerator x 9 2 + x 2 3 1 x x Eg. + 2 3

Partial Fractions with a quadratic factor Simple denominators Repeated factor

Partial fraction with simple denominators: + ax b An expression in the form can be split into partial ( q px + + )( ) rx s A + B + + fractions of the form . px q rx s 7 8 x Eg. Express in partial fractions. )( 1 2 ( x x ) 2 7 8 x x A B = + Let ) 2 2 ( 1 )( 2 1 2 x x x

7 8 x x A B = + ) 2 2 ( 1 )( 2 1 2 x x x + ) 1 7 8 x ( ) 2 x 2 ( x x A x B x = ) 2 ) 2 2 ( 1 )( 2 ( 1 )( x = ) 2 + ) 1 7 8 ( 2 ( B x A x x = + = 6 ) 0 ( A 3 , 2 B B Let x=2, = ) 2 + = 8 ( ( 2 1 ), 3 A A Let x= 0, 7 8 x 3 2 x = + So, the partial fractions are ) 2 2 ( 1 )( 2 1 2 x x x

Try: 13 6 x 2 1. Split into partial fractions. x x 2 3 12 + x x x 2. Split into partial fractions. )( 3 2 )( 1 ( + ) 3 x

Try: Answers 13 x 6 3 x 4 x = + 1 2 3 2 3 2 x x 12 x 3 + 8 1 x = + + 2 + + + ( 1 )( 2 3 )( ) 3 1 2 3 3 x x x x x

Partial fraction with a repeated factor: + + 2 ax bx c An expression in the form can be split ( + + 2 )( ) px q rx s into partial fractions of the form A + B + C + + + 2) ( px q rx s rx s 2 7 6 2 x 3 x 2 x x = + Eg. ) 3 ( x x 2 2 3 x

2 7 6 x x Express in partial fractions. ) 3 ( x x 2 Solution: 7 2 x x 2 6 x x A B C = + + 2 3 ( ) 3 x x x + + 2 2 7 x 6 ( ) 3 ( ) 3 x x A x Bx x Cx = 2 2 ( ) 3 ( ) 3 x x x + x = + 2 2 7 6 ( ) 3 ( ) 3 x x A x Bx Cx = = = = 2 18 6 12 9 ( 2 = , 2 C 3 C A Let x=3, x=0, x= 1, , ) 2 A + ) 2 + = ( ( 2 ), 3 B B 2 7 x 6 2 x 3 x 2 x x = + 2 2 3 ( ) 3 x x

Try: + 2 9 4 2 x Express in partial fractions. 2 ( ) 2 1 ( x + ) x

Answer: + 2 9 4 2 4 1 x = + + + 2 2 1 ( 2 ) 2 ( ) 1 ( 2 ) x x x x

Partial fraction with a quadratic factor: + + 2 ax bx c + An expression in the of the form , where + 2 ( )( ) px q rx s r and s have the same sign, can be split into partial fractions of the form + 2 A + Bx C + + px q rx s x + 5 6 2 2 + 2 x 1 x x = + Eg. + + + 2 2 ( 2 )( ) 4 4 x x

5 6 2+ x Express in partial fraction. ) 4 )( 2 ( + x x Solution: 5 + x x + 6 2 x A Bx C = + + + + 2 2 ( 2 )( x ) 4 4 Bx x x x + + + + 2 5 6 2 ( ) 4 x ( )( ) 2 A x C x = + + + + 2 ( 2 )( ) 4 ( 2 )( ) 4 x x = + + + + 2 5 6 ( ) 4 ( )( ) 2 x A x Bx C x = = , B C + Let x = -2, x = 0, x = 1, 16 6 1 8 8 10 , 2 A + A C ( = = = )( 2 + 1 3 = 1 ), 2 B x + 5 6 2 2 + 2 x 1 x x = + + + + 2 2 ( 2 )( ) 4 4 x x

Improper fractions: Degree of numerator = Degree of denominator: Eg. x + 3 4 + x x B Cx D = + + A + + + 2 2 1 ( 1 )( 2 ) 3 2 3 x x x Degree of numerator > degree of denominator: + 3 4 4 x x Eg. = + x + + 2 2 1 1 x x