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Factor x ( b 2 a ) + 6 ( b 2 a ) by pulling out the GCF.

( b 2 a ) ( x + 6 )

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Factoring a trinomial with leading coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial x 2 + 5 x + 6 has a GCF of 1, but it can be written as the product of the factors ( x + 2 ) and ( x + 3 ) .

Trinomials of the form x 2 + b x + c can be factored by finding two numbers with a product of c and a sum of b . The trinomial x 2 + 10 x + 16 , for example, can be factored using the numbers 2 and 8 because the product of those numbers is 16 and their sum is 10. The trinomial can be rewritten as the product of ( x + 2 ) and ( x + 8 ) .

Factoring a trinomial with leading coefficient 1

A trinomial of the form x 2 + b x + c can be written in factored form as ( x + p ) ( x + q ) where p q = c and p + q = b .

Can every trinomial be factored as a product of binomials?

No. Some polynomials cannot be factored. These polynomials are said to be prime.

Given a trinomial in the form x 2 + b x + c , factor it.

  1. List factors of c .
  2. Find p and q , a pair of factors of c with a sum of b .
  3. Write the factored expression ( x + p ) ( x + q ) .

Factoring a trinomial with leading coefficient 1

Factor x 2 + 2 x 15.

We have a trinomial with leading coefficient 1 , b = 2 , and c = −15. We need to find two numbers with a product of −15 and a sum of 2. In [link] , we list factors until we find a pair with the desired sum.

Factors of −15 Sum of Factors
1 , −15 −14
−1 , 15 14
3 , −5 −2
−3 , 5 2

Now that we have identified p and q as −3 and 5 , write the factored form as ( x 3 ) ( x + 5 ) .

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Does the order of the factors matter?

No. Multiplication is commutative, so the order of the factors does not matter.

Factor x 2 7 x + 6.

( x −6 ) ( x −1 )

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Factoring by grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping    by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial 2 x 2 + 5 x + 3 can be rewritten as ( 2 x + 3 ) ( x + 1 ) using this process. We begin by rewriting the original expression as 2 x 2 + 2 x + 3 x + 3 and then factor each portion of the expression to obtain 2 x ( x + 1 ) + 3 ( x + 1 ) . We then pull out the GCF of ( x + 1 ) to find the factored expression.

Factor by grouping

To factor a trinomial in the form a x 2 + b x + c by grouping, we find two numbers with a product of a c and a sum of b . We use these numbers to divide the x term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

Given a trinomial in the form a x 2 + b x + c , factor by grouping.
  1. List factors of a c .
  2. Find p and q , a pair of factors of a c with a sum of b .
  3. Rewrite the original expression as a x 2 + p x + q x + c .
  4. Pull out the GCF of a x 2 + p x .
  5. Pull out the GCF of q x + c .
  6. Factor out the GCF of the expression.

Factoring a trinomial by grouping

Factor 5 x 2 + 7 x 6 by grouping.

We have a trinomial with a = 5 , b = 7 , and c = −6. First, determine a c = −30. We need to find two numbers with a product of −30 and a sum of 7. In [link] , we list factors until we find a pair with the desired sum.

Factors of −30 Sum of Factors
1 , −30 −29
−1 , 30 29
2 , −15 −13
−2 , 15 13
3 , −10 −7
−3 , 10 7

So p = −3 and q = 10.

5 x 2 3 x + 10 x 6   Rewrite the original expression as  a x 2 + p x + q x + c . x ( 5 x 3 ) + 2 ( 5 x 3 ) Factor out the GCF of each part . ( 5 x 3 ) ( x + 2 ) Factor out the GCF  of the expression .
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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