# 4.2 Factoring the Greatest Common Factor of a Polynomial

When we study fractions, we learn that the **greatest common factor** (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex] The GCF of polynomials works the same way: [latex]4x[/latex] is the GCF of [latex]16x[/latex] and [latex]20x^2[/latex] because it is the largest polynomial that divides evenly into both [latex]16x[/latex] and [latex]20x^2[/latex].

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

## Greatest Common Factor

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

## How To

Given a polynomial expression, factor out the greatest common factor.

- Identify the GCF of the coefficients.
- Identify the GCF of the variables.
- Combine to find the GCF of the expression.
- Determine what the GCF needs to be multiplied by to obtain each term in the expression.
- Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

## Example 1: Factoring the Greatest Common Factor

- Factor [latex]6x^3y^3+45x^2y^2+21xy[/latex].

*Analysis*

After factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\left(3xy\right)\left(2x^2y^2+15xy+7\right)=6x^3y^3+45x^2y^2+21xy[/latex].

- Factor [latex]x\left(b^2-a\right)+6\left(b^2-a\right)[/latex] by pulling out the GCF.

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