Finding the Greatest Common Factor of a Polynomial
Learning Outcomes
- Factor the greatest common monomial out of a polynomial
Factor a Polynomial
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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+9 |
6x and 9 |
3 is a factor of 6x and 9 |
a2–2a |
a2 and −2a |
a is a factor of a2 and −2a |
4c3+4c |
4c3 and 4c |
4 and c are factors of 4c3 and 4c |
Remember that you can multiply a polynomial by a monomial as follows:
2(x+7)2⋅x+2⋅72x+14factorsproduct
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".
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,c are real numbers, then
a(b+c)=ab+ac and ab+ac=a(b+c)
Distributive Property Forward and Backward
Forward: Product of a number and a sum:
a(b+c)=a⋅b+a⋅c. You can say that “
a is being distributed over
b+c.”
Backward: Sum of the products:
a⋅b+a⋅c=a(b+c). 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+14
Solution
Step 1: Find the GCF of all the terms of the polynomial. |
Find the GCF of 2x and 14. |
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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 |
2x+14
2⋅x+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:
2(x+7)
2⋅x+2⋅7
2x+14✓ |
try it
[ohm_question]146330[/ohm_question]
Notice that in the example, we used the word factor as both a noun and a verb:
NounVerb7 is a factor of 14factor 2 from 2x+14
Factor the greatest common factor from a polynomial
- Find the GCF of all the terms of the polynomial.
- Rewrite each term as a product using the GCF.
- Use the Distributive Property ‘in reverse’ to factor the expression.
- 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+3
Answer:
Solution
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|
|
3a+3 |
Rewrite each term as a product using the GCF. |
3⋅a+3⋅1 |
Use the Distributive Property 'in reverse' to factor the GCF. |
3(a+1) |
Check by multiplying the factors to get the original polynomial. |
|
3(a+1)
3⋅a+3⋅1
3a+3✓ |
|
try it
[ohm_question]146331[/ohm_question]
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:
12x−60
Answer:
Solution
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|
|
12x−60 |
Rewrite each term as a product using the GCF. |
12⋅x−12⋅5 |
Factor the GCF. |
12(x−5) |
Check by multiplying the factors. |
|
12(x−5)
12⋅x−12⋅5
12x−60✓ |
|
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+9
Answer:
Solution
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|
|
3y2+6y+9 |
Rewrite each term as a product using the GCF. |
3⋅y2+3⋅2y+3⋅3 |
Factor the GCF. |
3(y2+2y+3) |
Check by multiplying. |
|
3(y2+2y+3)
3⋅y2+3⋅2y+3⋅3
3y2+6y+9✓ |
|
try it
[ohm_question]146333[/ohm_question]
In the next example, we factor a variable from a binomial.
example
Factor:
6x2+5x
Answer:
Solution
|
6x2+5x |
Find the GCF of 6x2 and 5x and the math that goes with it. |
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Rewrite each term as a product. |
x⋅6x+x⋅5 |
Factor the GCF. |
x(6x+5) |
Check by multiplying. |
|
x(6x+5)
x⋅6x+x⋅5
6x2+5x✓ |
|
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:
4x3−20x2
Answer:
Solution
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|
|
|
4x3−20x2 |
Rewrite each term. |
|
4x2⋅x−4x2⋅5 |
Factor the GCF. |
|
4x2(x−5) |
Check. |
4x2(x−5)
4x2⋅x−4x2⋅5
4x3−20x2✓
|
|
Example
Factor
25b3+10b2.
Answer: Find the GCF. From a previous example, you found the GCF of 25b3 and 10b2 to be 5b2.
25b3=5⋅5⋅b⋅b⋅b10b2=5⋅2⋅b⋅bGCF=5⋅b⋅b=5b2
Rewrite each term with the GCF as one factor.
25b3=5b2⋅5b10b2=5b2⋅2
Rewrite the polynomial using the factored terms in place of the original terms.
5b2(5b)+5b2(2)
Factor out the
5b2.
5b2(5b+2)
Answer
5b2(5b+2)
The factored form of the polynomial 25b3+10b2 is 5b2(5b+2). You can check this by doing the multiplication. 5b2(5b+2)=25b3+10b2.
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.
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+35y
Answer:
Solution
Find the GCF of 21y2 and 35y |
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|
|
|
21y2+35y |
Rewrite each term. |
|
7y⋅3y+7y⋅5 |
Factor the GCF. |
|
7y(3y+5) |
try it
[ohm_question]146338[/ohm_question]
example
Factor:
14x3+8x2−10x
Answer:
Solution
Previously, we found the GCF of 14x3,8x2,and10x to be 2x.
|
14x3+8x2−10x |
Rewrite each term using the GCF, 2x. |
2x⋅7x2+2x⋅4x−2x⋅5 |
Factor the GCF. |
2x(7x2+4x−5) |
2x(7x2+4x−5)
2x⋅7x2+2x⋅4x−2x⋅5
14x3+8x2−10x✓ |
|
try it
[ohm_question]146339[/ohm_question]
When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.
example
Factor:
−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 9y and 27 is 9. |
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Since the expression −9y−27 has a negative leading coefficient, we use −9 as the GCF. |
|
−9y−27 |
Rewrite each term using the GCF. |
−9⋅y+(−9)⋅3 |
Factor the GCF. |
−9(y+3) |
Check.
−9(y+3)
−9⋅y+(−9)⋅3
−9y−27✓ |
|
try it
[ohm_question]146340[/ohm_question]
Pay close attention to the signs of the terms in the next example.
example
Factor:
−4a2+16a
Answer:
Solution
The leading coefficient is negative, so the GCF will be negative. |
|
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Since the leading coefficient is negative, the GCF is negative, −4a. |
|
−4a2+16a |
Rewrite each term. |
−4a⋅a−(−4a)⋅4 |
Factor the GCF. |
−4a(a−4) |
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+45c2d2.
Answer: Factor 81c3d.
3⋅3⋅9⋅c⋅c⋅c⋅d
Factor
45c2d2.
3⋅3⋅5⋅c⋅c⋅d⋅d
Find the GCF.
3⋅3⋅c⋅c⋅d=9c2d
Rewrite each term as the product of the GCF and the remaining terms.
81c3d=9c2d(9c)45c2d2=9c2d(5d)
Rewrite the polynomial expression using the factored terms in place of the original terms.
9c2d(9c)+9c2d(5d)
Factor out
9c2d.
9c2d(9c+5d)
Answer
9c2d(9c+5d)
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 a⋅b+a⋅c form has been rewritten as a(b+c), where a is the GCF, the polynomial is in factored form.
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- Question ID 146341, 146340, 146339, 146338, 146337, 146335, 146333, 146331, 146330. Authored by: Lumen Learning. License: CC BY: Attribution.
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- Ex: Factor a Binomial - Greatest Common Factor (Basic). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
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