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Algebra 1

6.4.3 Factoring the GCF from Polynomials

Algebra 16.4.3 Factoring the GCF from Polynomials

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Activity

To factor a GCF from a polynomial and represent it in factored form, we must first find the GCF.

Let’s look at an example: 7z521sz37z521sz3.

Work individually on the following problems.

Factor the GCF from each polynomial. Remember that if the leading coefficient is negative, then the GCF will be negative.

1. 5x325x25x325x2

2. 8x3y10x2y2+12xy38x3y10x2y2+12xy3

3. 5y3+35y215y5y3+35y215y

4. 15x3y3x2y2+6xy315x3y3x2y2+6xy3

5. 4p3q12p2q2+16pq24p3q12p2q2+16pq2

6. 5x(x+1)+3(x+1)5x(x+1)+3(x+1)

7. 3b(b2)13(b2)3b(b2)13(b2)

8. 6m(m5)7(m5)6m(m5)7(m5)

Self Check

Factor: 8 a 3 b + 2 a 2 b 2 6 a b 3 .
  1. 2 a b ( 4 a 2 + a b 3 b 2 )
  2. 2 b ( 4 a 3 + a 2 b 3 a b 2 )
  3. a b ( 8 a 2 + 2 a b 6 b 2 )
  4. 2 a b ( 4 a 2 + a b 3 b 2 )

Additional Resources

Factoring the GCF from Polynomials

It is sometimes useful to represent a number as a product of factors, for example, 12 as 2·62·6 or 3·43·4. In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as 3x2+15x3x2+15x, and end with its factors, 3x(x+5)3x(x+5). To do this, we apply the Distributive Property “in reverse.” We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

DISTRIBUTIVE PROPERTY

If aa, bb, and cc are real numbers, then

a(b+c)=ab+aca(b+c)=ab+ac and ab+ac=a(b+c)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 you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

Example 1

Factor: 8m312m2n+20mn28m312m2n+20mn2.

How to factor the greatest common factor from a polynomial:

Step 1 - Find the GCF of all the terms of the polynomial. For this problem, find the GCF of 8m38m3, 12m2n12m2n, and 20mn220mn2.

Factorization of 8m³, 12m²n, and 20mn² showing shared factors 2 and m circled in red and the greatest common factor (GCF) calculated as 2 × 2 × m = 4m.

Step 2 - Rewrite each term as a product using the GCF. In this case, rewrite 8m38m3, 12m2n12m2n, and 20mn220mn2 as products of their GCF, 4m4m.

  • 8m3=4m·2m28m3=4m·2m2
  • 12m2n=4m·3mn12m2n=4m·3mn
  • 20mn2=4m·5n220mn2=4m·5n2

8m312m2n+20mn28m312m2n+20mn2

4m·2m24m·3mn+4m·5n24m·2m24m·3mn+4m·5n2

Step 3 - Use the “reverse” Distributive Property to factor the expression.

4m(2m23mn+5n2)4m(2m23mn+5n2)

Step 4 - Check by multiplying the factors.

4m(2m23mn+5n2)4m(2m23mn+5n2)

4m·2m24m·3mn+4m·5n24m·2m24m·3mn+4m·5n2

8m312m2n+20mn28m312m2n+20mn2

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

Example 2

Factor: 4a3+36a28a4a3+36a28a.

The leading coefficient is negative, so the GCF will be negative.

Step 1 - Rewrite each term using the GCF, 4a4a.

4a·a2(4a)·9a+(4a)·24a·a2(4a)·9a+(4a)·2

Step 2 - Factor the GCF.

4a(a29a+2)4a(a29a+2)

Step 3 - Check.

4a(a29a+2)4a(a29a+2)

4a·a2(4a)·9a+(4a)·24a·a2(4a)·9a+(4a)·2

4a3+36a28a4a3+36a28a

So far, our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

Example 3

Factor: 3y(y+7)4(y+7)3y(y+7)4(y+7).

The GCF is the binomial y+7y+7.

3y(y+7)4(y+7)3y(y+7)4(y+7)

Step 1 - Factor the GCF, (y+7)(y+7).

(y+7)(3y4)(y+7)(3y4)

Step 2 - Check on your own by multiplying.

Try it

Try It: Factoring the GCF from Polynomials

Factor.

1. 9xy2+6x2y2+21y39xy2+6x2y2+21y3

2. 4b3+16b28b4b3+16b28b

3. 4m(m+3)7(m+3)4m(m+3)7(m+3)

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