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1.5 Factoring polynomials  (Page 2/6)

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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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