3.3 GCD and LCM of Polynomials

. GCD and LCM of Polynomials . . Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of Polynomials In our previous class we have learnt how to find the GCD (HCF) of second degree and third degree expressions by the method of factorization. Now we shall learn how to find the GCD of the given polynomials by the method of long division. As discussed in Chapter , (Numbers and Sequences) to find GCD of two positive integers using Euclidean Algorithm, similar techniques can be employed for two given polynomials also. The following procedure gives a systematic way of finding Greatest Common Divisor of two given polynomials f x ( ) and g x ( ) . Step : First, divide f ( x ) by g x ( ) to obtain f x g x q x r x ( ) ( ) where q x ( ) is the quotient and r x ( ) is the remainder. Then, deg deg r x g x  <   Step : If the remainder r x ( ) is non-zero, divide g x ( ) by r x ( ) to obtain g x r x q x r x ( ) ( ) where r x ( ) is the new remainder. Then deg deg r x r x   <   . If the remainder r x ( ) is zero, then r x ( ) is the required GCD. Step : If r x ( ) is non-zero, then continue the process until we get zero as remainder. The divisor at this stage will be the required GCD. We write GCD f x g x ( ), ( )   to denote the GCD of the polynomials f x g x ( ), ( ).