Easy Algebra Step-by-Step

Factoring Polynomials 121

monomial factor. The greatest common monomial factor is the product of

the greatest common numerical factor and a second component made up of

the common variable factors, each with the highest power common to each

term. You can refer to the greatest common monomial factor as the greatest

common factor (GCF).

Problem Find the GCF for the terms in the polynomial 12x^8 y^3 − 8x^6 y^7 z^2.

Solution

Step 1. Find the numerical factor of the GCF by fi nding the greatest com-

mon numerical factor of 12 and 8.

The factors of 12 are 1, 2, 3, 4 , 6, and 12, and the factors of 8 are 1,

2, 4 , and 8. The numerical factor of the GCF is 4.

Step 2. Identify the common variable factors, each with the highest power

common to x^8 y^3 and x^6 y^7 z^2.

x and y are the common variable factors. The highest power of x

that is common to each term is x^6 , and the highest power of y that is

common to each term is y^3. The common variable component of the

GCF is x^6 y^3.

Step 3. Write the GCF as the product of the results of steps 1 and 2.

The GCF for the terms in the polynomial 12x^8 y^3 − 8 x^6 y^7 z^2 is 4x^6 y^3.

Problem Factor.

a. 12 x^8 y^3 − 8 x^6 y^7 z^2

b. 15 x^2 − 3 x

c. x^3 y − xy + y

d. 4 x + 4 y

Solution

a. 12 x^8 y^3 − 8 x^6 y^7 z^2

Step 1. Determine the GCF for 12x^8 y^3 and 8x^6 y^7 z^2.

GCF = 4 x^6 y^3

Step 2. Rewrite each term of the polynomial as an equivalent product of

4 x^6 y^3 and a second factor.

12 8

x^83 y − xy^67 z^2

When factoring out the

GCF, check your work by

mentally multiplying the

factors of your answers.