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. 5 3 6 9 369 = (5 00) 69 3 0 0 0 6 9 Notice the is in the hundreds column so it s really 00. M Patel (April 0) t. Machar Academy
69 is not less than 5, so we continue, 5 3 6 9 369 = (5 00) 69 3 0 0 0 6 9 69 = (5 0) 9 6 0 0 9 9 is not less than 5, so we continue, 6 5 3 6 9 3 69 = (5 00) 69 3 0 0 0 6 9 69 = (5 0) 9 6 0 0 9 9 = (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, 3 69 5 = 6 5 M Patel (April 0) t. Machar Academy
Eample Divide 3 5 by. 3 5 3 5 = ( ) 7 5 7 5 As 7 5 has a degree which is not less than the degree of, we continue, 7 3 5 3 5 = ( ) 7 5 7 5 7 5 = ( ) 7 9 7 9 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 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
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
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 = 5 7 0 Hence, 3 5 ( )( )( 3) = ( ) 5( ) 7 0( 3) M Patel (April 0) 5 t. Machar Academy
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
Linear Factor and Irreducible Quadratic Factor Eample 5 Find partial fractions for (3 )( ). (3 )( ) = 3 T U = ( ) ( T U ) (3 ) = gives, 3 9 = 9 0 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