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Study Guides > ALGEBRA / TRIG I

Finding the Greatest Common Factor of a Polynomial

Learning Outcomes

  • Factor the greatest common monomial out of a polynomial
Factor a Polynomial
Beetles pinned to a surface as a collection with a mini volkswagen beetle car in the mix. One of these things is not like the others.
Before we solve polynomial equations, we will practice finding the greatest common factor of a polynomial. If you can find common factors for each term of a polynomial, then you can factor it, and solving will be easier.     To help you practice finding common factors, identify factors that the terms of the polynomial have in common in the table below.
Polynomial Terms Common Factors
6x+96x+9 6x6x and 99 33 is a factor of 6x6x and  99
a22aa^{2}–2a a2a^{2} and 2a−2a a is a factor of a2a^{2} and 2a−2a
4c3+4c4c^{3}+4c 4c34c^{3} and 4c4c 44 and c are factors of 4c34c^{3} and  4c4c
Remember that you can multiply a polynomial by a monomial as follows:

2(x+7)factors2x+272x+14product\begin{array}{ccc}\hfill 2\left(x + 7\right)&\text{factors}\hfill \\ \hfill 2\cdot x + 2\cdot 7\hfill \\ \hfill 2x + 14&\text{product}\hfill \end{array}

Here, we will start with a product, like 2x+142x+14, and end with its factors, 2(x+7)2\left(x+7\right). To do this we apply the Distributive Property "in reverse". To factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in a factored form. Recall that the distributive property of multiplication over addition states that a product of a number and a sum is the same as the sum of the products.

Distributive Property

If a,b,ca,b,c are real numbers, then a(b+c)=ab+ac and ab+ac=a(b+c)a\left(b+c\right)=ab+ac\text{ and }ab+ac=a\left(b+c\right)

Distributive Property Forward and Backward

Forward: Product of a number and a sum: a(b+c)=ab+aca\left(b+c\right)=a\cdot{b}+a\cdot{c}. You can say that “aa is being distributed over b+cb+c.” Backward: Sum of the products: ab+ac=a(b+c)a\cdot{b}+a\cdot{c}=a\left(b+c\right). Here you can say that “a is being factored out.”
We first learned that we could distribute a factor over a sum or difference, now we are learning that we can "undo" the distributive property with factoring. 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

Factor: 2x+142x+14 Solution
Step 1: Find the GCF of all the terms of the polynomial. Find the GCF of 2x2x and 1414. .
Step 2: Rewrite each term as a product using the GCF. Rewrite 2x2x and 1414 as products of their GCF, 22. 2x=2x2x=2\cdot x 14=2714=2\cdot 7 2x+142x+14 2x+27\color{red}{2}\cdot x+\color{red}{2}\cdot7
Step 3: Use the Distributive Property 'in reverse' to factor the expression. 2(x+7)2\left(x+7\right)
Step 4: Check by multiplying the factors. Check: 2(x+7)2(x+7) 2x+272\cdot{x}+2\cdot{7} 2x+142x+14\quad\checkmark
 

try it

[ohm_question]146330[/ohm_question]
Notice that in the example, we used the word factor as both a noun and a verb:

Noun7 is a factor of 14Verbfactor 2 from 2x+14\begin{array}{cccc}\text{Noun}\hfill & & & 7\text{ is a factor of }14\hfill \\ \text{Verb}\hfill & & & \text{factor }2\text{ from }2x+14\hfill \end{array}

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.
Notice in the next example how, when we factor 3 out of the expression, we are left with a factor of 1.

example

Factor: 3a+33a+3

Answer: Solution

.
3a+33a+3
Rewrite each term as a product using the GCF. 3a+31\color{red}{3}\cdot a+\color{red}{3}\cdot1
Use the Distributive Property 'in reverse' to factor the GCF. 3(a+1)3(a+1)
Check by multiplying the factors to get the original polynomial.
3(a+1)3(a+1) 3a+313\cdot{a}+3\cdot{1} 3a+33a+3\quad\checkmark

 

try it

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

example

Factor: 12x6012x - 60

Answer: Solution

.
12x6012x-60
Rewrite each term as a product using the GCF. 12x125\color{red}{12}\cdot x-\color{red}{12}\cdot 5
Factor the GCF. 12(x5)12(x-5)
Check by multiplying the factors.
12(x5)12(x-5) 12x12512\cdot{x}-12\cdot{5} 12x6012x-60\quad\checkmark

 

try it

[ohm_question]146332[/ohm_question]
Watch the following video to see more examples of factoring the GCF from a binomial. https://youtu.be/68M_AJNpAu4 Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

example

Factor: 3y2+6y+93{y}^{2}+6y+9

Answer: Solution

.
3y2+6y+93y^2+6y+9
Rewrite each term as a product using the GCF. 3y2+32y+33\color{red}{3}\cdot{y}^{2}+\color{red}{3}\cdot 2y+\color{red}{3}\cdot 3
Factor the GCF. 3(y2+2y+3)3(y^{2}+2y+3)
Check by multiplying.
3(y2+2y+3)3(y^{2}+2y+3) 3y2+32y+333\cdot{y^2}+3\cdot{2y}+3\cdot{3} 3y2+6y+93y^{2}+6y+9\quad\checkmark

 

try it

[ohm_question]146333[/ohm_question]
In the next example, we factor a variable from a binomial.

example

Factor: 6x2+5x6{x}^{2}+5x

Answer: Solution

6x2+5x6{x}^{2}+5x
Find the GCF of 6x26{x}^{2} and 5x5x and the math that goes with it. .
Rewrite each term as a product. x6x+x5\color{red}{x}\cdot{6x}+\color{red}{x}\cdot{5}
Factor the GCF. x(6x+5)x\left(6x+5\right)
Check by multiplying.
x(6x+5)x\left(6x+5\right) x6x+x5x\cdot 6x+x\cdot 5 6x2+5x6{x}^{2}+5x\quad\checkmark

 

try it

[ohm_question]146335[/ohm_question]
When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!

example

Factor: 4x320x24{x}^{3}-20{x}^{2}

Answer: Solution

.
4x320x24x^3-20x^2
Rewrite each term. 4x2x4x25\color{red}{4{x}^{2}}\cdot x - \color{red}{4{x}^{2}}\cdot 5
Factor the GCF. 4x2(x5)4x^2(x-5)
Check. 4x2(x5)4x^2(x-5) 4x2x4x254x^2\cdot{x}-4x^2\cdot{5} 4x320x24x^3-20x^2\quad\checkmark  

Example

Factor 25b3+10b225b^{3}+10b^{2}.

Answer: Find the GCF. From a previous example, you found the GCF of 25b325b^{3} and 10b210b^{2} to be 5b25b^{2}.

  25b3=55bbb  10b2=52bbGCF=5bb=5b2\begin{array}{l}\,\,25b^{3}=5\cdot5\cdot{b}\cdot{b}\cdot{b}\\\,\,10b^{2}=5\cdot2\cdot{b}\cdot{b}\\\text{GCF}=5\cdot{b}\cdot{b}=5b^{2}\end{array}

Rewrite each term with the GCF as one factor.

25b3=5b25b10b2=5b22\begin{array}{l}25b^{3} = 5b^{2}\cdot5b\\10b^{2}=5b^{2}\cdot2\end{array}

Rewrite the polynomial using the factored terms in place of the original terms.

5b2(5b)+5b2(2)5b^{2}\left(5b\right)+5b^{2}\left(2\right)

Factor out the 5b25b^{2}.

5b2(5b+2)5b^{2}\left(5b+2\right)

Answer

5b2(5b+2)5b^{2}\left(5b+2\right)

The factored form of the polynomial 25b3+10b225b^{3}+10b^{2} is 5b2(5b+2)5b^{2}\left(5b+2\right). You can check this by doing the multiplication. 5b2(5b+2)=25b3+10b25b^{2}\left(5b+2\right)=25b^{3}+10b^{2}. Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over. For example:

25b3+10b2=5(5b3+2b2)           Factor out 5.                             =5b2(5b+2)               Factor out b2.\begin{array}{l}25b^{3}+10b^{2}=5\left(5b^{3}+2b^{2}\right)\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }5.\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5b^{2}\left(5b+2\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }b^{2}.\end{array}

Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.

try it

[ohm_question]146337[/ohm_question]
 

example

Factor: 21y2+35y21{y}^{2}+35y

Answer: Solution

Find the GCF of 21y221{y}^{2} and 35y35y .
21y2+35y21y^2+35y
Rewrite each term. 7y3y+7y5\color{red}{7y}\cdot 3y + \color{red}{7y}\cdot 5
Factor the GCF. 7y(3y+5)7y(3y+5)

 

try it

[ohm_question]146338[/ohm_question]
 

example

Factor: 14x3+8x210x14{x}^{3}+8{x}^{2}-10x

Answer: Solution Previously, we found the GCF of 14x3,8x2,and10x14{x}^{3},8{x}^{2},\text{and}10x to be 2x2x.

14x3+8x210x14{x}^{3}+8{x}^{2}-10x
Rewrite each term using the GCF, 2x. 2x7x2+2x4x2x5\color{red}{2x}\cdot 7{x}^{2}+\color{red}{2x}\cdot4x-\color{red}{2x}\cdot 5
Factor the GCF. 2x(7x2+4x5)2x\left(7{x}^{2}+4x - 5\right)
2x(7x2+4x5)2x(7x^2+4x-5) 2x7x2+2x4x2x52x\cdot{7x^2}+2x\cdot{4x}-2x\cdot{5} 14x3+8x210x14x^3+8x^2-10x\quad\checkmark

 

try it

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When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.

example

Factor: 9y27-9y - 27

Answer: Solution

When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of 9y9y and 2727 is 99. .
Since the expression 9y27−9y−27 has a negative leading coefficient, we use 9−9 as the GCF.
9y27-9y - 27
Rewrite each term using the GCF. 9y+(9)3\color{red}{-9}\cdot y + \color{red}{(-9)}\cdot 3
Factor the GCF. 9(y+3)-9\left(y+3\right)
Check. 9(y+3)-9(y+3) 9y+(9)3-9\cdot{y}+(-9)\cdot{3} 9y27-9y-27\quad\checkmark

 

try it

[ohm_question]146340[/ohm_question]
Pay close attention to the signs of the terms in the next example.

example

Factor: 4a2+16a-4{a}^{2}+16a

Answer: Solution

The leading coefficient is negative, so the GCF will be negative.
.
Since the leading coefficient is negative, the GCF is negative, 4a−4a.
4a2+16a-4{a}^{2}+16a
Rewrite each term. 4aa(4a)4\color{red}{-4a}\cdot{a}-\color{red}{(-4a)}\cdot{4}
Factor the GCF. 4a(a4)-4a\left(a - 4\right)
Check on your own by multiplying.

 

TRY IT

[ohm_question]146341[/ohm_question]
This next example shows factoring a binomial when there are two different variables in the expression.

Example

Factor 81c3d+45c2d281c^{3}d+45c^{2}d^{2}.

Answer: Factor 81c3d81c^{3}d.

339cccd3\cdot3\cdot9\cdot{c}\cdot{c}\cdot{c}\cdot{d}

Factor 45c2d245c^{2}d^{2}.

335ccdd3\cdot3\cdot5\cdot{c}\cdot{c}\cdot{d}\cdot{d}

Find the GCF.

33ccd=9c2d3\cdot3\cdot{c}\cdot{c}\cdot{d}=9c^{2}d

Rewrite each term as the product of the GCF and the remaining terms.

  81c3d=9c2d(9c)45c2d2=9c2d(5d)\begin{array}{l}\,\,81c^{3}d=9c^{2}d\left(9c\right)\\45c^{2}d^{2}=9c^{2}d\left(5d\right)\end{array}

Rewrite the polynomial expression using the factored terms in place of the original terms.

9c2d(9c)+9c2d(5d)9c^{2}d\left(9c\right)+9c^{2}d\left(5d\right)

Factor out 9c2d9c^{2}d.

9c2d(9c+5d)9c^{2}d\left(9c+5d\right)

Answer

9c2d(9c+5d)9c^{2}d\left(9c+5d\right)

The following video provides two more examples of finding the greatest common factor of a binomial https://youtu.be/25_f_mVab_4 This last example shows finding the greatest common factors of trinomials. https://youtu.be/3f1RFTIw2Ng

Summary

A whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in ab+aca\cdot{b}+a\cdot{c} form has been rewritten as a(b+c)a\left(b+c\right), where a is the GCF, the polynomial is in factored form.

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