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Introduction to Factoring Polynomials

Find the greatest common factor of two or more expressions

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

In The Language of Algebra we factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

First we will find the greatest common factor of two numbers.

Example

Try it.

Find the greatest common factor of \(24\) and \(36.\)

Solution

Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form.	Factor 24 and 36.
Step 2: List all factors--matching common factors in a column.
In each column, circle the common factors.	Circle the 2, 2, and 3 that are shared by both numbers.
Step 3: Bring down the common factors that all expressions share.	Bring down the 2, 2, 3 and then multiply.
Step 4: Multiply the factors.		The GCF of 24 and 36 is 12.

Notice that since the GCF is a factor of both numbers, \(24\) and \(36\) can be written as multiples of \(12.\)

\[\begin{array}{l}24=12\cdot 2 \\ 36=12\cdot 3\end{array}\]

In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.

Example

Try it.

Find the greatest common factor of \(5x\ \text{and}\ 15.\)

Solution

Factor each number into primes. Circle the common factors in each column. Bring down the common factors.
	The GCF of 5x and 15 is 5.

In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

Condensed — the full section is in OpenStax Prealgebra 2e.

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, \(12\) as \(2\cdot 6\ \text{or}\ 3\cdot 4\text{),}\) in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

\[\begin{array}{lll}2(x & + & 7)\ \text{factors} \\ 2\cdot x & + & 2\cdot 7 \\ 2x & + & 14\ \text{product}\end{array}\]

Here, we will start with a product, like \(2x+14,\) and end with its factors, \(2(x+7).\) To do this we apply the Distributive Property “in reverse”.

The form on the left is used to multiply. The form on the right is used to factor.

So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!

Example

Try it.

Factor: \(2x+14.\)

Solution

Step 1: Find the GCF of all the terms of the polynomial.	Find the GCF of 2x and 14.
Step 2: Rewrite each term as a product using the GCF.	Rewrite 2x and 14 as products of their GCF, 2. \(2x=2⋅x\) \(14=2⋅7\)
Step 3: Use the Distributive Property 'in reverse' to factor the expression.		\(2(x+7)\)
Step 4: Check by multiplying the factors.		Check:

Notice that in , we used the word factor as both a noun and a verb:

\[\begin{array}{llll}\text{Noun} & & & 7\ \text{is a factor of}\ 14 \\ \text{Verb} & & & \text{factor}\ 2\ \text{from}\ 2x+14\end{array}\]

Example

Try it.

Factor: \(3a+3.\)

Solution



Rewrite each term as a product using the GCF.
Use the Distributive Property 'in reverse' to factor the GCF.
Check by multiplying the factors to get the original polynomial.

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Example

Try it.

Factor: \(12x-60.\)

Solution



Rewrite each term as a product using the GCF.
Factor the GCF.
Check by multiplying the factors.

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Find the greatest common factor.
1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Bring down the common factors that all expressions share.
4. Multiply the factors.
Distributive Property
- If \(a\), \(b\), \(c\) are real numbers, then
  \(a(b+c)=ab+ac\) and \(ab+ac=a(b+c)\)
Factor the greatest common factor from a polynomial.
1. Find the GCF of all the terms of the polynomial.
2. Rewrite each term as a product using the GCF.
3. Use the Distributive Property ‘in reverse’ to factor the expression.
4. Check by multiplying the factors.

Chapter Practice Test

In the following exercises, simplify each expression.

In the following exercises, factor the greatest common factor from each polynomial.

In the following exercises, simplify, and write your answer in decimal form.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Factor \(56\) into primes.
If you missed this problem, review .

Afslør svaret

\(2⋅2⋅2⋅7\)
Multiply: \(-3(6a+11).\)
If you missed this problem, review .

Afslør svaret

\(-18a-33\)
Multiply: \(4{x}^{2}({x}^{2}+3x-1).\)
If you missed this problem, review .

Afslør svaret

\(4{x}^{4}+12{x}^{3}-4{x}^{2}\)

Find the greatest common factor of \(24\) and \(36.\)

Afslør svaret

Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form.	Factor 24 and 36.
Step 2: List all factors--matching common factors in a column.
In each column, circle the common factors.	Circle the 2, 2, and 3 that are shared by both numbers.
Step 3: Bring down the common factors that all expressions share.	Bring down the 2, 2, 3 and then multiply.
Step 4: Multiply the factors.		The GCF of 24 and 36 is 12.

Notice that since the GCF is a factor of both numbers, \(24\) and \(36\) can be written as multiples of \(12.\)

\[\begin{array}{l}24=12\cdot 2 \\ 36=12\cdot 3\end{array}\]

Find the greatest common factor: \(54,36.\)

Afslør svaret

18
Find the greatest common factor: \(48,80.\)

Afslør svaret

16
Find the greatest common factor of \(5x\ \text{and}\ 15.\)

Afslør svaret

Factor each number into primes.
Circle the common factors in each column.
Bring down the common factors.
The GCF of 5x and 15 is 5.
Find the greatest common factor: \(7y,\ 14.\)

Afslør svaret

7
Find the greatest common factor: \(22,\ 11m.\)

Afslør svaret

11

Find the greatest common factor of \(12{x}^{2}\) and \(18{x}^{3}.\)

Afslør svaret

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors.
	\(\text{The GCF of}\ 12{x}^{2}\ \text{and}\ 18{x}^{3}\ \text{is}\ 6{x}^{2}\)

Find the greatest common factor: \(16{x}^{2},\ 24{x}^{3}.\)

Afslør svaret

8x²
Find the greatest common factor: \(27{y}^{3},\ 18{y}^{4}.\)

Afslør svaret

9y³

Find the greatest common factor of \(14{x}^{3},\ 8{x}^{2},\ 10x.\)

Afslør svaret

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors.
	\(\text{The GCF of}\ 14{x}^{3}\ \text{and}\ 8{x}^{2}\text{, and}10x\ \text{is}\ 2x\)

Find the greatest common factor: \(21{x}^{3},\ 9{x}^{2},\ 15x.\)

Afslør svaret

3x
Find the greatest common factor: \(25{m}^{4},\ 35{m}^{3},\ 20{m}^{2}.\)

Afslør svaret

5m²

Factor: \(2x+14.\)

Afslør svaret

Step 1: Find the GCF of all the terms of the polynomial.	Find the GCF of 2x and 14.
Step 2: Rewrite each term as a product using the GCF.	Rewrite 2x and 14 as products of their GCF, 2. \(2x=2⋅x\) \(14=2⋅7\)
Step 3: Use the Distributive Property 'in reverse' to factor the expression.		\(2(x+7)\)
Step 4: Check by multiplying the factors.		Check:

Factor: \(4x+12.\)

Afslør svaret

4(x + 3)
Factor: \(6a+24.\)

Afslør svaret

6(a + 4)
Factor: \(3a+3.\)

Afslør svaret

Rewrite each term as a product using the GCF.
Use the Distributive Property 'in reverse' to factor the GCF.
Check by multiplying the factors to get the original polynomial.
Factor: \(9a+9.\)

Afslør svaret

9(a + 1)
Factor: \(11x+11.\)

Afslør svaret

11(x + 1)
Factor: \(12x-60.\)

Afslør svaret

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by multiplying the factors.
Factor: \(11x-44.\)

Afslør svaret

11(x − 4)
Factor: \(13y-52.\)

Afslør svaret

13(y − 4)
Factor: \(3{y}^{2}+6y+9.\)

Afslør svaret

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by multiplying.
Factor: \(4{y}^{2}+8y+12.\)

Afslør svaret

4(y² + 2y + 3)
Factor: \(6{x}^{2}+42x-12.\)

Afslør svaret

6(x² + 7x − 2)

Factor: \(6{x}^{2}+5x.\)

Afslør svaret

	\(6{x}^{2}+5x\)
Find the GCF of \(6{x}^{2}\) and \(5x\) and the math that goes with it.
Rewrite each term as a product.
Factor the GCF.	\(x(6x+5)\)
Check by multiplying.
\(x(6x+5)\) \(x⋅6x+x⋅5\) \(6{x}^{2}+5x✓\)

Factor: \(9{x}^{2}+7x.\)

Afslør svaret

x(9x + 7)
Factor: \(5{a}^{2}-12a.\)

Afslør svaret

a(5a − 12)
Factor: \(4{x}^{3}-20{x}^{2}.\)

Afslør svaret

Rewrite each term.
Factor the GCF.
Check.
Factor: \(2{x}^{3}+12{x}^{2}.\)

Afslør svaret

2x²(x + 6)
Factor: \(6{y}^{3}-15{y}^{2}.\)

Afslør svaret

3y²(2y − 5)
Factor: \(21{y}^{2}+35y.\)

Afslør svaret

Find the GCF of \(21{y}^{2}\) and \(35y\)
Rewrite each term.
Factor the GCF.
Factor: \(18{y}^{2}+63y.\)

Afslør svaret

9y(2y + 7)
Factor: \(32{k}^{2}+56k.\)

Afslør svaret

8k(4k + 7)
Factor: \(14{x}^{3}+8{x}^{2}-10x.\)

Afslør svaret

Previously, we found the GCF of \(14{x}^{3},\ 8{x}^{2},\ \text{and}\ 10x\) to be \(2x.\)
\(14{x}^{3}+8{x}^{2}-10x\)
Rewrite each term using the GCF, 2x.
Factor the GCF. \(2x(7{x}^{2}+4x-5)\)
Factor: \(18{y}^{3}-6{y}^{2}-24y.\)

Afslør svaret

6y(3y² − y − 4)
Factor: \(16{x}^{3}+8{x}^{2}-12x.\)

Afslør svaret

4x(4x² + 2x − 3)

Factor: \(-9y-27.\)

Afslør svaret

When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of 9y and 27 is 9.
Since the expression −9y−27 has a negative leading coefficient, we use −9 as the GCF.
	\(-9y-27\)
Rewrite each term using the GCF.
Factor the GCF.	\(-9(y+3)\)

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Introduction to Factoring Polynomials

Find the greatest common factor of two or more expressions
Factor the greatest common factor from a polynomial
Factor each coefficient into primes. Write all variables with exponents in expanded form.
List all factors—matching common factors in a column. In each column, circle the common factors.
Bring down the common factors that all expressions share.
Multiply the factors.
Find the GCF of all the terms of the polynomial.
Rewrite each term as a product using the GCF.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Mere i Arithmetic

Order of operations Fractions Percentages Powers and roots

Find the GCF of \(21{y}^{2}\) and \(35y\)

Rewrite each term.
Factor the GCF.

	\(14{x}^{3}+8{x}^{2}-10x\)
Rewrite each term using the GCF, 2x.
Factor the GCF.	\(2x(7{x}^{2}+4x-5)\)