Partial Fractions. Prerequisites: Solving simple equations; comparing coefficients; factorising simple quadratics and cubics; polynomial division.

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1 Prerequisites: olving simple equations; comparing coefficients; factorising simple quadratics and cubics; polynomial division. Maths Applications: Integration; graph sketching. Real-World Applications: Population growth. Rational Functions Definition: A polynomial divided by another polynomial is called a rational function, p () q () If deg p < deg q, the above is called a proper rational function, whereas if deg p deg q, the above is called an improper rational function. A rational function can be written in terms of a proper rational function. Polynomial Long Division Polynomial long division is a technique used to write an improper rational function in terms of a proper rational function. The technique can be illustrated by analogy with long division of whole numbers. Eample Divide 3 69 by = (5 00) Notice the is in the hundreds column so it s really 00. M Patel (April 0) t. Machar Academy

2 69 is not less than 5, so we continue, = (5 00) = (5 0) is not less than 5, so we continue, = (5 00) = (5 0) = (5 6) 9 0 As is less than 5, we stop. Putting the 3 calculations on the RH together gives, 3 69 = (5 00) (5 0) (5 6) In other words, 3 69 = (5 6) The quotient is 6 and the remainder is. This can be written in the alternative form, = 6 5 M Patel (April 0) t. Machar Academy

3 Eample Divide 3 5 by = ( ) As 7 5 has a degree which is not less than the degree of, we continue, = ( ) = ( ) As the degree of 9 is less than the degree of, we stop. Putting the two calculations on the RH together gives, upon simplifying, 3 5 = ( ) ( ) = ( ) ( 7) 9 o, the quotient is 7 and the remainder is 9. This can be written in the alternative form, 3 5 = ( 7) 9 Reducible and Irreducible polynomials ome polynomials can be factorised using the real number system, others can t. M Patel (April 0) 3 t. Machar Academy

4 Definition: A polynomial p is reducible if it can be factorised into polynomials, none of which are equal to the polynomials and p. A polynomial is irreducible if it is not reducible. Partial Fraction Decompositions Rational functions can be broken down into proper rational functions. Theorem (Partial Fraction Decomposition Theorem): Any rational function p q can be written as a polynomial plus a sum of proper rational functions each of which is of the form, g () r () n (n N ) where r is an irreducible factor of q and deg g < deg r ; such proper rational functions are called partial fractions of p q. Rational functions that have at most a cubic denominator will be studied. The cubic will factorise into either 3 linear factors ( of which may be the same) or linear and irreducible quadratic factor. Types of Arising from a Cubic Denominator In the following table, a 0, each factor is non-zero and, T R. Factor a b (non-repeated linear) ( a b) (repeated linear) a b c (irreducible quadratic) Partial Fraction a b T a b ( a b) T a b c M Patel (April 0) t. Machar Academy

5 Long division can be done by dividing with a quadratic, cubic etc. For eample, check (by epanding the denominator and then long dividing) that, 3 ( )( ) = ( )( ) 3 Distinct Linear Factors Eample 3 Find partial fractions for 3 5. ( )( )( 3) 3 5 ( )( )( 3) = T U 3 Multiply both sides of the above equation by ( ) ( ) ( 3) to give, 3 5 = ( )( 3) T ( )( 3) U ( )( ) = gives, 3( ) 5 = ( )( 3) 0 0 = ()( ) = imilarly, = and = 3 respectively give, T =, U = Hence, 3 5 ( )( )( 3) = ( ) 5( ) 7 0( 3) M Patel (April 0) 5 t. Machar Academy

6 Distinct Linear Factors (with of these Repeated) Eample Epress 6 ( ) ( ) in partial fractions. 6 ( ) ( ) = T ( ) U 6 = ( )( ) T ( ) U ( ) = gives, = 0 T ( 6) 0 6T = T = = gives, 6 = 0 0 U (36) 36U = 36 U = Comparing coefficients of gives, = U = 0 o, 6 ( ) ( ) = ( ) M Patel (April 0) 6 t. Machar Academy

7 Linear Factor and Irreducible Quadratic Factor Eample 5 Find partial fractions for (3 )( ). (3 )( ) = 3 T U = ( ) ( T U ) (3 ) = gives, 3 9 = = 3 9 = 3 3 Epanding out the RH of the main equation gives, = ( 3T ) (T 3U ) ( U ) Comparing coefficients of both sides gives, 3T = T = 5 3 U = U = 0 3 Hence, (3 )( ) = 3 3(3 ) 5 0 3( ) M Patel (April 0) 7 t. Machar Academy

INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS. Introduction It is possible to integrate any rational function, constructed as the ratio of polynomials by epressing it as a sum of simpler fractions