GREATEST COMMON FACTOR OF POLYNOMIALS

Th e Greatest Common Factor, abbreviated as GCF, of two or more polynomials is a polynomial, of the highest common possible degree, that is a factor of the given two or more polynomials. 

This concept is similar to the greatest common divisor of two integers.

For example, Consider the expressions 14xy² and 42xy.

The common divisors of 14 and 42 are 2, 7 and 14.

Their GCF is thus 14. The only common divisors of xy² and xy are x, y and xy; their GCF is thus xy.

14xy²  =  2 ⋅ 7 ⋅ x ⋅ y ⋅ y

42xy  =  2 ⋅ 3 ⋅ 7 ⋅ x ⋅ y

Therefore, the required GCF of 14xy² and 42xy is

=  2 ⋅ 7 ⋅ x ⋅ y

=  14xy 

Find GCF of Polynomials by Factoring 

(i)  Each expression is to be resolved into factors first.

(ii) The product of factors having the highest common powers in those factors will be the GCF.

(iii) If the expression have numerical coefficient, find their GCF separately and then prefix it as a coefficient to the GCF for the given expressions.

Example 1 : 

Find the greatest common factor of the following : 

16x³y²   and   24xy³z

Solution :

16x³y²  =  2⁴ ⋅ x³ ⋅ y²  =   2³ ⋅ 2 ⋅ x² ⋅ x ⋅ y²

  24xy³z  =  2³ ⋅ 3 ⋅ x ⋅ y³ ⋅ z  =  2³ ⋅ 3 ⋅ x ⋅ y² ⋅ y ⋅ z  

So, GCF is

=  2³ ⋅ x ⋅ y²

=  8xy²

Example 2 : 

Find the greatest common factor of the following : 

(y³ + 1)   and   (y³ - 1)

Solution :

Algebraic Identities : 

a³ + b³  =  (a + b)(a² - ab + b²)

a² - b²  =  (a + b)(a - b)

Using the above identities, factor the given polynomials and find the greatest common factor. 

y³ + 1  =  y³ + 1³  =  (y + 1)(y² - y + 1)

y² - 1  =  y² - 1²  =  (y + 1)(y - 1)

So, GCF is

=  (y + 1)

Example 3 : 

Find the greatest common factor of the following : 

(2x² - 18)   and   (x² - 2x - 3)

Solution :

Factor the given polynomials and find the greatest common factor. 

Factor (2x² - 18) : 

2x² - 18  =  2(x² - 9)

2x² - 18  =  2(x² - 3²)

2x² - 18  =  2(x + 3)(x - 3) -----(1)

Factor (x² - 2x - 3) : 

x² - 2x - 3  =  x² - 3x + x - 3

x² - 2x - 3  =  x(x - 3) + 1(x - 3)

x² - 2x - 3  =  (x - 3)(x + 1) -----(2)

From (1) and (2), GCF is 

(x - 3)

Example 4 : 

Find the greatest common factor of the following : 

(a - b)², (b - c)³, (c - a)⁴

Solution :

For the given polynomials, there is no common factor other than one. 

So, GCF is 1. 

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