Find the GCF (Greatest Common Factor) for the following group of polynomials:?

Question

Can you solve and explain?

A) 7yx^3, 21x^3y

B) (x - y), (x^4 - y^4), (x^2 - y^2)

Answer 1

Factor each polynomial completely, and then take each factor the minimum number of times that it occurs in either polynomial.

A) 7yx^3 = 7*y*x*x*x
21x^3 y = 3*7*x*x*x*y

3 occurs 0 or 1 times, so it's included 0 times in the GCF.
7 occurs 1 time in each, so it's included 1 time in the GCF.
x occurs 3 times in each, so it's included 3 times in the GCF.
y occurs 1 time in each, so it's included once in the GCF.

GCF = 7x^3 y

B) (x-y) is factored already.
(x^4-y^4) = (x^2-y^2)(x^2+y^2) = (x+y)(x-y)(x^2+y^2)
(x^2-y^2) = (x+y)(x-y).

(x-y) occurs once as a factor in each polynomial, but no other factors occur at least once in every polynomial, so they do not belong to the GCF. Therefore the GCF is x-y.