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.

\(2x(3y-5)\)

**\(\longrightarrow\)**

Distributive Property

\(6xy-10x\)

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.

\(2x(3y-5)\)

**\(\longleftarrow\)**

Reverse Distributive Property

\(6xy-10x\)

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.

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.

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 \(144x^{3}+60x^{2}+30x\)

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 \(36x^{3}y^{2}-24x^{2}y^{3}+72xy^{4}\).

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.

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 \(4x^{2}y-6x^{3}y^{3}+8x^{4}y^{2}\)

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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