Finding the Greatest Common Factor of a Polynomial

Learning Outcomes

  • 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, 1212 as 26 or 3426 or 34) 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:

2(x+7)factors2x+272x+14product2(x+7)factors2x+272x+14product

Here, we will start with a product, like 2x+142x+14, and end with its factors, 2(x+7)2(x+7). To do this we apply the Distributive Property “in reverse”.

Distributive Property

If a,b,ca,b,c are real numbers, then

a(b+c)=ab+ac and ab+ac=a(b+c)a(b+c)=ab+ac and ab+ac=a(b+c)

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

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=2x

14=2714=27

2x+142x+14

2x+272x+27

Step 3: Use the Distributive Property ‘in reverse’ to factor the expression. 2(x+7)2(x+7)
Step 4: Check by multiplying the factors. Check:

2(x+7)2(x+7)

2x+272x+27

2x+142x+14

 

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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+14Noun7 is a factor of 14Verbfactor 2 from 2x+14

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.

 

example

Factor: 3a+33a+3

 

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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

Factor: 12x6012x60

 

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Watch the following video to see more examples of factoring the GCF from a binomial.

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+93y2+6y+9

 

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In the next example, we factor a variable from a binomial.

example

Factor: 6x2+5x6x2+5x

 

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

example

Factor: 4x320x24x320x2

 

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example

Factor: 21y2+35y21y2+35y

 

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example

Factor: 14x3+8x210x14x3+8x210x

 

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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: 9y279y27

 

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Pay close attention to the signs of the terms in the next example.

example

Factor: 4a2+16a4a2+16a

 

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