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Greatest Common Divisor
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Greatest Common Divisor (G.C.D.)
Greatest Common Factor (GCF) of two numbers is the largest number that divides both of the given numbers exactly. It is called their Greatest Common Divisor (G.C.D.)

Follow 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 powers of primes.
Setp IV: Identify common irreducible divisors and find the smallest 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. If there is no common divisor, the GCD is 1.
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Example: Find the g.c.d. of the following polynomials.
p(x) = 60(3x4 + x3 - 2x2) and q(x) = 10(x6 + 3x5 + 2x4)
Solution: Given
p(x) = 60(3x4 + x3- 2x2)
= 22 * 3 * 5 * x2(3x2 + x - 2)
= 22 * 3 * 5 * x2(3x2 + 3x - 2x - 2)
= 22 * 3 * 5 * x2[3x(x + 1) - 2(x + 1)]
= 22 * 3 * 5 * x2 * (x + 1)(3x - 2)
q(x)= 10(x6+ 3x5+ 2x4)
= 2 * 5 * x4(x2 + 3x + 2)
= 2 * 5 * x4(x2 + 2x + x + 2)
= 2 * 5 * x4[x(x + 2) + 1(x + 2)]
= 2 * 5 * x4(x + 1)(x + 2)
Here, common irreducible divisors are 2, 5, x and (x + 1). There least exponents are 1, 1, 2 and 1 respectively.
Therefore,
                 G.C.D. = 2 * 5 * x2 * (x + 1) = 10x2(x + 1).
 
   
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