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Mathematics LibreTexts

6.5: General Strategy for Factoring Polynomial Expressions

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Learning Objectives

By the end of this section, you will be able to:

  • Recognize and use the appropriate method to factor a polynomial completely

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

You have now become acquainted with all the methods of factoring that you will need in this course. The following chart summarizes all the factoring methods we have covered, and outlines a strategy you should use when factoring polynomials.

GENERAL STRATEGY FOR FACTORING POLYNOMIALS

This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.

USE A GENERAL STRATEGY FOR FACTORING POLYNOMIALS.
  1. Is there a greatest common factor?
    Factor it out.
  2. Is the polynomial a binomial, trinomial, or are there more than three terms?
    If it is a binomial:
    • Is it a sum?
      Of squares? Sums of squares do not factor.
      Of cubes? Use the sum of cubes pattern.
    • Is it a difference?
      Of squares? Factor as the product of conjugates.
      Of cubes? Use the difference of cubes pattern.
    If it is a trinomial:
    • Is it of the form x2+bx+c? Undo FOIL.
    • Is it of the form ax2+bx+c?
      If a and c are squares, check if it fits the trinomial square pattern.
      Use the trial and error or “ac” method.
    If it has more than three terms:
    • Use the grouping method.
  3. Check.
    Is it factored completely?
    Do the factors multiply back to the original polynomial?

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Example 6.5.1

Factor completely: 7x321x270x.

Solution

7x321x270xIs there a GCF? Yes, 7x.Factor out the GCF.7x(x23x10)In the parentheses, is it a binomial, trinomial,or are there more terms?Trinomial with leading coefficient 1.“Undo” FOIL.7x(x)(x)7x(x+2)(x5)Is the expression factored completely? Yes.Neither binomial can be factored.Check your answer.Multiply.7x(x+2)(x5)7x(x25x+2x10)7x(x23x10)7x321x270x

Try It 6.5.1

Factor completely: 8y3+16y224y.

Answer

8y(y1)(y+3)

Try It 6.5.2

Factor completely: 5y315y2270y.

Answer

5y(y9)(y+6)

Be careful when you are asked to factor a binomial as there are several options!

Example 6.5.2

Factor completely: 24y2150

Solution

24y2150Is there a GCF? Yes, 6.Factor out the GCF.6(4y225)In the parentheses, is it a binomial, trinomialor are there more than three terms? Binomial.Is it a sum? No.Is it a difference? Of squares or cubes? Yes, squares.6((2y)2(5)2)Write as a product of conjugates.6(2y5)(2y+5)Is the expression factored completely?Neither binomial can be factored.Check:Multiply.6(2y5)(2y+5)6(4y225)24y2150

Try It 6.5.3

Factor completely: 16x336x.

Answer

4x(2x3)(2x+3)

Try It 6.5.4

Factor completely: 27y248.

Answer

3(3y4)(3y+4)

The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.

Example 6.5.3

Factor completely: 4a212ab+9b2.

Solution

4a212ab+9b2Is there a GCF? No.Is it a binomial, trinomial, or are there more terms?Trinomial with a1. But the first term is a perfect square.Is the last term a perfect square? Yes.(2a)212ab+(3b)2Does it fit the pattern, a22ab+b2? Yes.(2a)212ab+(3b)22(2a)(3b)Write it as a square.(2a3b)2Is the expression factored completely? Yes.The binomial cannot be factored.Check your answer.Multiply.(2a3b)2(2a)22·2a·3b+(3b)24a212ab+9b2

Try It 6.5.5

Factor completely: 4x2+20xy+25y2.

Answer

(2x+5y)2

Try It 6.5.6

Factor completely: 9x224xy+16y2.

Answer

(3x4y)2

Remember, sums of squares do not factor, but sums of cubes do!

Example 6.5.4

Factor completely 12x3y2+75xy2.

Solution

12x3y2+75xy2Is there a GCF? Yes, 3xy2.Factor out the GCF.3xy2(4x2+25)In the parentheses, is it a binomial, trinomial, or arethere more than three terms? Binomial.Is it a sum? Of squares? Yes.Sums of squares are prime.Is the expression factored completely? Yes.Check:Multiply.3xy2(4x2+25)12x3y2+75xy2

Try It 6.5.7

Factor completely: 50x3y+72xy.

Answer

2xy(25x2+36)

Try It 6.5.8

Factor completely: 27xy3+48xy.

Answer

3xy(9y2+16)

When using the sum or difference of cubes pattern, being careful with the signs.

Example 6.5.5

Factor completely: 24x3+81y3.

Solution

Is there a GCF? Yes, 3. .
Factor it out. .
In the parentheses, is it a binomial, trinomial,
of are there more than three terms? Binomial.
 
Is it a sum or difference? Sum.  
Of squares or cubes? Sum of cubes. .
Write it using the sum of cubes pattern. .
Is the expression factored completely? Yes. .
Check by multiplying.  
Try It 6.5.9

Factor completely: 250m3+432n3.

Answer

2(5m+6n)(25m230mn+36n2)

Try It 6.5.10

Factor completely: 2p3+54q3.

Answer

2(p+3q)(p23pq+9q2)

Example 6.5.6

Factor completely: 3x5y48xy.

Solution

3x5y48xyIs there a GCF? Factor out 3xy3xy(x416)Is the binomial a sum or difference? Of squares or cubes?Write it as a difference of squares.3xy((x2)2(4)2)Factor it as a product of conjugates3xy(x24)(x2+4)The first binomial is again a difference of squares.3xy((x)2(2)2)(x2+4)Factor it as a product of conjugates.3xy(x2)(x+2)(x2+4)Is the expression factored completely? Yes.Check your answer.Multiply.3xy(x2)(x+2)(x2+4)3xy(x24)(x2+4)3xy(x416)3x5y48xy

Try It 6.5.11

Factor completely: 4a5b64ab.

Answer

4ab(a2+4)(a2)(a+2)

Try It 6.5.12

Factor completely: 7xy57xy.

Answer

7xy(y2+1)(y1)(y+1)

Example 6.5.7

Factor completely: 4x2+8bx4ax8ab.

Solution

4x2+8bx4ax8abIs there a GCF? Factor out the GCF, 4.4(x2+2bxax2ab)There are four terms. Use grouping.4[x(x+2b)a(x+2b)]4(x+2b)(xa)Is the expression factored completely? Yes.Check your answer.Multiply.4(x+2b)(xa)4(x2ax+2bx2ab)4x2+8bx4ax8ab

Try It 6.5.13

Factor completely: 6x212xc+6bx12bc.

Answer

6(x+b)(x2c)

Try It 6.5.14

Factor completely: 16x2+24xy4x6y.

Answer

2(4x1)(2x+3y)

Taking out the complete GCF in the first step will always make your work easier.

Example 6.5.8

Factor completely: 40x2y+44xy24y.

Solution

40x2y+44xy24yIs there a GCF? Factor out the GCF, 4y.4y(10x2+11x6)Factor the trinomial with a1.4y(10x2+11x6)4y(5x2)(2x+3)Is the expression factored completely? Yes.Check your answer.Multiply.4y(5x2)(2x+3)4y(10x2+11x6)40x2y+44xy24y

Try It 6.5.15

Factor completely: 4p2q16pq+12q.

Answer

4q(p3)(p1)

Try It 6.5.16

Factor completely: 6pq29pq6p.

Answer

3p(2q+1)(q2)

When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.

Example 6.5.9

Factor completely: 9x212xy+4y249.

Solution

9x212xy+4y249Is there a GCF? No.With more than 3 terms, use grouping. Last 2 termshave no GCF. Try grouping first 3 terms.9x212xy+4y249Factor the trinomial with a1. But the first term is aperfect square.Is the last term of the trinomial a perfect square? Yes.(3x)212xy+(2y)249Does the trinomial fit the pattern, a22ab+b2? Yes.(3x)212xy+(2y)2492(3x)(2y))Write the trinomial as a square.(3x2y)249Is this binomial a sum or difference? Of squares orcubes? Write it as a difference of squares.(3x2y)272Write it as a product of conjugates.((3x2y)7)((3x2y)+7)(3x2y7)(3x2y+7)Is the expression factored completely? Yes.Check your answer.Multiply.(3x2y7)(3x2y+7)9x26xy21x6xy+4y2+14y+21x14y499x212xy+4y249

Try It 6.5.17

Factor completely: 4x212xy+9y225.

Answer

(2x3y5)(2x3y+5)

Try It 6.5.18

Factor completely: 16x224xy+9y264.

Answer

(4x3y8)(4x3y+8)

Key Concepts

This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.

  • How to use a general strategy for factoring polynomials.
    1. Is there a greatest common factor?
      Factor it out.
    2. Is the polynomial a binomial, trinomial, or are there more than three terms?
      If it is a binomial:
      Is it a sum?
      Of squares? Sums of squares do not factor.
      Of cubes? Use the sum of cubes pattern.
      Is it a difference?
      Of squares? Factor as the product of conjugates.
      Of cubes? Use the difference of cubes pattern.
      If it is a trinomial:
      Is it of the form x2+bx+c? Undo FOIL.
      Is it of the form ax2+bx+c?
      If a and c are squares, check if it fits the trinomial square pattern.
      Use the trial and error or “ac” method.
      If it has more than three terms:
      Use the grouping method.
    3. Check.
      Is it factored completely?
      Do the factors multiply back to the original polynomial?

This page titled 6.5: General Strategy for Factoring Polynomial Expressions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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