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Factoring
Greatest Common Factor for a polynomial

The Greatest Common Factor, or GCF, for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial.

Find the Greatest Common Factor for the polynomial 4x5+12x2

The terms of the polynomial are 4x5 and 12x2. The largest number that divides the coefficients is 4, and the highest power of x that is a factor of x5 and x2 is x2. Therefore, the greatest common factor for 4x5+12x2 is 4x2; that is 4x2 is the largest monomial that divides each term of 4x5+12x2.
We ussually write:

4x5+12x2=4x2(x3+3)

Factor 3x(x+6)-10(x+6)
The GCF is (x+6).
The factoring form is (x+6)(3x-10)

The greatest common factor, or GCF, of 2 or more expressions is the greatest factor that divides evenly into both expressions.

Find the greatest common factor of:

a) 24 and 60
b) 18x2 and 27x3

Solution:

a)
24: 1, 2, 3, 4, 6, 8, 12, 24
60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
GCF(24,60)=12

b) 9x2