GREATEST COMMON DIVISOR (GCD) OR HIGHEST COMMON FACTOR (HCF) The greatest common divisor (g.c.d.) of two polynomials p(x) and q(x) is that common divisor which has highest degree among all common divisors and in which the coefficient of highest degree term is positive.
The g.c.d. of polynomials can be obtained by using the following algorithm.
ALGORITHM
STEP I Obtain the polynomials. Let the polynomials be p(x) and q(x).
Step II Factorise the polynomials p(x) and q(x).
STEP III Express p(x) and q(x) as a product of powers of irreducible factors. Also, express the numerical factor as a product of pozvers of primes.
STEP iv Identify common irreducible divisors and find the smallest (least) exponents of these common divisors in the given polynomials.
STEP V Raise the common irreducible.divisors to the smallest exponents found in step IV and multiply them to get the GCD (HCF). If there is no common divisor, the GCD (HCF) is 1.
Example : Find the g.c.d. of the following polynomials:
SOLUTION We have,
We observe that the common irreducible divisors of p (x) and q (x) are 2 and (x — 2). The least exponent of these divisors are 2 and 1 respectively.
\( GCD = 2^2 (x – 2)^1 = 4 (x-2) \)