How to Factor Out the Greatest Common Factor of Polynomials
Factoring out the greatest common factor of a polynomial is essentially the opposite of multiplying a monomial by a polynomial.
When you multiply a monomial by a polynomial, you use the distributive property to convert the polynomial from factored form to expanded form.
Factored Form
\(2x(3y-5)\)
\(\longrightarrow\)
Distributive Property
Expanded Form
\(6xy-10x\)
When you factor out the greatest common factor of a polynomial, you use the reverse distributive property to convert the polynomial from expanded form to factored form.
Factored Form
\(2x(3y-5)\)
\(\longleftarrow\)
Reverse Distributive Property
Expanded Form
\(6xy-10x\)
What is the "Greatest Common Factor"?
Learn more about the GCF of numbers.
The greatest common factor (or GCF) of a set of numbers is the largest number that all of the numbers in the set can be evenly divided by.
The greatest common factor of a polynomial is the expression that all of the terms in the polynomial can be evenly divided by.
Finding the GCF of a polynomial is useful because the GCF is the expression that will be outside of the parentheses when we re-write an expanded polynomial in factored form.
How to Find the
Greatest Common Factor
of Polynomials
The process for finding the greatest common factor of a polynomial is very similar to the process for finding the GCF of a number.
You can find the GCF in your head by asking, "What can all of the terms of this polynomial be divided by?"
Or you can use the prime factorization method...
- Find the prime factorization of the numbers in each term and write the exponents in expanded form.
- Find the factors that all of the terms have in common.
- Multiply all the common factors together.
I recommend using the prime factorization method when you are first learning how to find the greatest common factor of a polynomial.
After you've practiced several problems with the prime factorization method, you'll be able to see patterns in the polynomials that will make it really easy to find the GCF in your head.
Easy Example:
Find the GCF of \(144x^{3}+60x^{2}+30x\)
Step 1: Find the prime factorization of the numbers in each term and write the exponents in expanded form.
\((2)(2)(2)(2)(3)(3)xxx+(2)(2)(3)(5)xx+(2)(3)(5)x\)
Step 2: Find the factors that all of the terms have in common.
\(\color{black}{(2)}(2)(2)(2)\color{black}{(3)}(3)\color{black}{x}xx+\color{black}{(2)(3)}(2)(5)\color{black}{x}x+\color{black}{(2)(3)}(5)\color{black}{x}\)
Step 3: Multiply all the common factors together.
\(\color{black}{(2)(3)x}=6x\)
Answer: \(6x\) is the GCF of \(144x^{3}+60x^{2}+30x\)
Harder Example:
Find the GCF of \(36x^{3}y^{2}-24x^{2}y^{3}+72xy^{4}\)
Step 1: Find the prime factorization of the numbers in each term and write the exponents in expanded form.
\((2)(2)(3)(3)xxxyy-(2)(2)(2)(3)xxyyy+(2)(2)(2)(3)(3)xyyyy\)
Step 2: Find the factors that all of the terms have in common.
\(\color{black}{(2)(2)(3)}(3)\color{black}{x}xx\color{black}{yy}-\color{black}{(2)(2)}(2)\color{black}{(3)x}x\color{black}{yy}y+\color{black}{(2)(2)}(2)\color{black}{(3)}(3)\color{black}{xyy}yy\)
Step 3: Multiply all the common factors together.
\(\color{black}{(2)(2)(3)xyy} = 12xy^{2}\)
Answer: \(12xy^{2}\) is the GCF of \(36x^{3}y^{2}-24x^{2}y^{3}+72xy^{4}\).
How to Factor Out the
Greatest Common Factor
of a Polynomial
To reverse the process of multiplying a monomial by a polynomial, you have to divide each term in the polynomial by the GCF (which is a monomial).
To factor out the greatest common factor of a polynomial...
- Find the greatest common factor of the polynomial.
- Factor out the greatest common factor from each term.
- Write the polynomial in factored form.
Easy Example:
Factor out the GCF of \(12x^{4}-16x^{3}+28x^{2}\)
Step 1: Find the greatest common factor of the polynomial.
You can do this in your head by noticing that each of the terms in the polynomial can be divided by 4 and \(x^{2}\).
However, I'll write out the prime factorization method to show you how it works.
The prime factors of 12 are 2, 2, 3 and \(x^{4}\) can be expanded to xxxx.
The prime factors of 16 are 2, 2, 2, 2 and \(x^{3}\) can be expanded to xxx.
The prime factors of 28 are 2, 2, 7 and \(x^{2}\) can be expanded to xx.
\(\color{black}{(2)(2)}(3)\color{black}{xx}xx-\color{black}{(2)(2)}(2)(2)\color{black}{xx}x+\color{black}{(2)(2)}(7)\color{black}{xx}\)
All the terms have two 2's and two x's in common, so the GCF of the polynomial is \(\color{black}{(2)(2)xx}=4x^{2}\).
Step 2: Factor out the greatest common factor from each term.
\(\color{black}{4x^{2}}(3x^{2})-\color{black}{4x^{2}}(4x)+\color{black}{4x^{2}}(7)\)
Step 3: Write the polynomial in factored form.
The GCF will be written outside of parentheses and the left-over factors from each term will be written inside parentheses.
We are allowed to re-write polynomials like this because of the reverse distributive property.
\(\color{black}{4x^{2}y}(3x^{2}-4x+7)\)
Answer: \(2x^{2}y(2-3xy^{2}+4x^{2}y)\) is the factored form of \(4x^{2}y-6x^{3}y^{3}+8x^{4}y^{2}\)
Harder Example:
Factor out the GCF of \(4x^{2}y-6x^{3}y^{3}+8x^{4}y^{2}\)
Step 1: Find the greatest common factor of the polynomial.
\(\color{black}{(2)}(2)\color{black}{xxy}-\color{black}{(2)}(3)\color{black}{xx}x\color{black}{y}yy+\color{black}{(2)}(2)(2)\color{black}{xx}xx\color{black}{y}y\)
The GCF of the polynomial is \(\color{black}{2x^{2}y}\).
Step 2: Factor out the greatest common factor from each term.
\(\color{black}{2x^{2}y}(2)-\color{black}{2x^{2}y}(3xy^{2})+\color{black}{2x^{2}y}(4x^{2}y)\)
Step 3: Write the polynomial in factored form.
\(\color{black}{2x^{2}y}(2-3xy^{2}+4x^{2}y)\)
Answer: \(2x^{2}y(2-3xy^{2}+4x^{2}y)\) is the factored form of \(4x^{2}y-6x^{3}y^{3}+8x^{4}y^{2}\)
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