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7.5: General Strategy for Factoring Polynomials

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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
Note

Before you get started, take this readiness quiz.

  1. Factor y22y24.
    If you missed this problem, review Example 7.2.19.
  2. Factor 3t2+17t+10.
    If you missed this problem, review Example 7.3.28.
  3. Factor 36p260p+25.
    If you missed this problem, review Example 7.4.1.
  4. Factor 5x280.
    If you missed this problem, review Example 7.4.31.

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. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. Figure 7.5.1 outlines a strategy you should use when factoring polynomials.

This figure presents a general strategy for factoring polynomials. First, at the top, there is GCF, which is where factoring starts. Below this, there are three options, binomial, trinomial, and more than three terms. For binomial, there are the difference of two squares, the sum of squares, the sum of cubes, and the difference of cubes. For trinomials, there are two forms, x squared plus bx plus c and ax squared 2 plus b x plus c. There are also the sum and difference of two squares formulas as well as the “a c” method. Finally, for more than three terms, the method is grouping.
Figure 7.5.1
FACTOR 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 aa and cc 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 7.5.1

Factor completely: 4x5+12x4

Solution

 Is there a GCF?  Yes, 4x44x5+12x4 Factor out the GCF. 4x4(x+3) In the parentheses, is it a binomial, a  trinomial, or are there more than three terms?  Binomial.  Is it a sum?  Yes.  Of squares? Of cubes?  No.  Check.  Is the expression factored completely?  Yes. Multiply. 4x4(x+3)4x4x+4x434x5+12x4

Try It 7.5.2

Factor completely: 3a4+18a3

Answer

3a3(a+6)

Try It 7.5.3

Factor completely: 45b6+27b5

Answer

9b5(5b+3)

Example 7.5.4

Factor completely: 12x211x+2

Solution

    .
Is there a GCF? No.  
Is it a binomial, trinomial, or are
there more than three terms?
Trinomial.  
Are a and c perfect squares? No, a = 12,
not a perfect square.
 
Use trial and error or the “ac” method.
We will use trial and error here.
  .
This table has the heading of 12 x squared minus 11 x plus 2 and gives the possible factors. The first column is labeled possible factors and the second column is labeled product. Four rows have not an option in the product column. This is explained by the text, “if the trinomial has no common factors, then neither factor can contain a common factor”. The last factors, 3 x - 2 in parentheses and 4 x - 1 in parentheses, give the product of 12 x squared minus 11 x plus 2. Check. (3x2)(4x1)12x23x8x+212x211x+2
Try It 7.5.5

Factor completely: 10a217a+6

Answer

(5a6)(2a1)

Try It 7.5.6

Factor completely: 8x218x+9

Answer

(2x3)(4x3)

Example 7.5.7

Factor completely: g3+25g

Solution

 Is there a GCF? Yes, g.g3+25g Factor out the GCF. g(g2+25) In the parentheses, is it a binomial, trinomial,  or are there more than three terms?  Binomial.  Is it a sum? Of squares?  Yes.  Sums of squares are prime.  Check.  Is the expression factored completely?  Yes.  Multiply. g(g2+25)g3+25g

Try It 7.5.8

Factor completely: x3+36x

Answer

x(x2+36)

Try It 7.5.9

Factor completely: 27y2+48

Answer

3(9y2+16)

Example 7.5.10

Factor completely: 12y275

Solution

 Is there a GCF? Yes, 3.12y275 Factor out the GCF. 3(4y225) In the parentheses, is it a binomial, trinomial,  or are there more than three terms?  Binomial.  Is it a sum? No.  Is it a difference? Of squares or cubes?  Yes, squares. 3((2y)2(5)2) Write as a product of conjugates. 3(2y5)(2y+5) Check.  Is the expression factored completely?  Yes. Neither binomial is a difference of  squares.  Multiply.3(2y5)(2y+5)3(4y225)12y275

Try It 7.5.11

Factor completely: 16x336x

Answer

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

Try It 7.5.12

Factor completely: 27y248

Answer

3(3y4)(3y+4)

Example 7.5.13

Factor completely: 4a212ab+9b2

Solution

Is 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. .
Does it fit the pattern, a22ab+b2? Yes. .
Write it as a square.   .
Check your answer.    
Is the expression factored completely?    
  Yes.    
  The binomial is not a difference of squares.    
  Multiply.    
(2a3b)2    
(2a)222a3b+(3b)2    
4a212ab+9b2
Try It 7.5.14

Factor completely: 4x2+20xy+25y2

Answer

(2x+5y)2

Try It 7.5.15

Factor completely: 9m2+42mn+49n2

Answer

(3m+7n)2

Example 7.5.16

Factor completely: 6y218y60

Solution

 Is there a GCF? Yes, 6.6y218y60 Factor out the GCF.  Trinomial with leading coefficient 16(y23y10) In the parentheses, is it a binomial, trinomial,  or are there more terms?  "Undo' FOIL. 6(y)(y)6(y+2)(y5) Check your answer.  Is the expression factored completely?  Yes. Neither binomial is a difference of squares.  Multiply. 6(y+2)(y5)6(y25y+2y10)6(y23y10)6y218y60

Try It 7.5.17

Factor completely: 8y2+16y24

Answer

8(y1)(y+3)

Try It 7.5.18

Factor completely: 5u215u270

Answer

5(u9)(u+6)

Example 7.5.19

Factor completely: 24x3+81

Solution

Is there a GCF? Yes, 3. 24x3+81
Factor it out.   3(8x3+27)
In the parentheses, is it a binomial, trinomial,
or 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. 3(2x+3)(4x26x+9)
Check by multiplying.   We leave the check to you.
Try It 7.5.20

Factor completely: 250m3+432

Answer

2(5m+6)(25m230m+36)

Try It 7.5.21

Factor completely: 81q3+192

Answer

3(3q+4)(9q212q+16)

Example 7.5.22

Factor completely: 2x432

Solution

 Is there a GCF? Yes, 2.2x432 Factor out the GCF. 2(x416) In the parentheses, is it a binomial, trinomial,  or are there more than three terms?  Binomial.  Is it a sum or difference?  Yes.  Of squares or cubes?  Difference of squares. 2((x2)2(4)2) Write it as a product of conjugates. 2(x24)(x2+4) The first binomial is again a difference of squares. 2((x)2(2)2)(x2+4) Write it as a product of conjugates. 2(x2)(x+2)(x2+4) Is the expression factored completely?  Yes.  None of these binomials is a difference of squares.  Check your answer.  Multiply. 2(x2)(x+2)(x2+4)2(x2)(x+2)(x2+4)2(x10)2x432

Try It 7.5.23

Factor completely: 4a464

Answer

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

Try It 7.5.24

Factor completely: 7y47

Answer

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

Example 7.5.25

Factor completely: 3x2+6bx3ax6ab

Solution

 Is there a GCF? Yes, 3.3x2+6bx3ax6ab Factor out the GCF. 3(x2+2bxax2ab) In the parentheses, is it a binomial, trinomial,  More than 3 or are there more terms?  terms.  Use grouping. 3[x(x+2b)a(x+2b)]3(x+2b)(xa) Check your answer.  Is the expression factored completely? Yes.  Multiply. 3(x+2b)(xa)3(x2ax+2bx2ab)3x23ax+6bx6ab

Try It 7.5.26

Factor completely: 6x212xc+6bx12bc

Answer

6(x+b)(x2c)

Try It 7.5.27

Factor completely: 16x2+24xy4x6y

Answer

2(4x1)(x+3y)

Example 7.5.28

Factor completely: 10x234x24

Solution

 Is there a GCF? Yes, 2.10x234x24 Factor out the GCF. 2(5x217x12) In the parentheses, is it a binomial, trinomial,  Trinomial with  or are there more than three terms?  a1 Use trial and error or the "ac" method. 2(5x217x12)2(5x+3)(x4) Check your answer. Is the expression factored  completely? Yes.  Multiply. 2(5x+3)(x4)2(5x220x+3x12)2(5x217x12)10x234x24

Try It 7.5.29

Factor completely: 4p216p+12

Answer

4(p1)(p3)

Try It 7.5.30

Factor completely: 6q29q6

Answer

3(q2)(2q+1)

Key Concepts

  • General Strategy for Factoring Polynomials See Figure 7.5.1.
  • How to Factor 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 7.5: General Strategy for Factoring Polynomials 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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