General Strategy for Factoring Polynomials

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

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.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_06_04_002_img.jpg)
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!

Factor completely: 7x321x270x.

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

Factor completely: 8y3+16y224y.

8y(y1)(y+3)

Factor completely: 5y315y2270y.

5y(y9)(y+6)

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

Factor completely: 24y2150.

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

Factor completely: 16x336x.

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

Factor completely: 27y248.

3(3y4)(3y+4)

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

Factor completely: 4a212ab+9b2.

4a212ab+9b2Is there a GCF? No.Is it a binomial, trinomial, or are there more terms?Trinomial witha1.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)2−12ab+−2(2a)(3b)(3b)2Write 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

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

(2x+5y)2

Factor completely: 9x224xy+16y2.

(3x4y)2

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

Factor completely 12x3y2+75xy2.

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

Factor completely: 50x3y+72xy.

2xy(25x2+36)

Factor completely: 27xy3+48xy.

3xy(9y2+16)

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

Factor completely: 24x3+81y3.

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.

Factor completely: 250m3+432n3.

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

Factor completely: 2p3+54q3.

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

Factor completely: 3x5y48xy.

3x5y48xyIs there a GCF? Factor out3xy3xy(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

Factor completely: 4a5b64ab.

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

Factor completely: 7xy57xy.

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

Factor completely: 4x2+8bx4ax8ab.

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

Factor completely: 6x212xc+6bx12bc.

6(x+b)(x2c)

Factor completely: 16x2+24xy4x6y.

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

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

Factor completely: 40x2y+44xy24y.

40x2y+44xy24yIs there a GCF? Factor out the GCF,4y.4y(10x2+11x6)Factor the trinomial witha1.4y(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

Factor completely: 4p2q16pq+12q.

4q(p3)(p1)

Factor completely: 6pq29pq6p.

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.

Factor completely: 9x212xy+4y249.

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 witha1.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)2−12xy+−2(3x)(2y)(2y)249Write 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

Factor completely: 4x212xy+9y225.

(2x3y5)(2x3y+5)

Factor completely: 16x224xy+9y264.

(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?

Practice Makes Perfect

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

2n2+13n7
(2n1)(n+7)
8x29x3
a5+9a3
a3(a2+9)
75m3+12m
121r2s2
(11rs)(11r+s)
49b236a2
8m232
8(m2)(m+2)
36q2100
25w260w+36
(5w6)2
49b2112b+64
m2+14mn+49n2
(m+7n)2
64x2+16xy+y2
7b2+7b42
7(b+3)(b-2)
30n2+30n+72
3x4y81xy
3(x3)(x2+3x+9)
4x5y32x2y
k416
(k2)(k+2)(k2+4)
m481
5x5y280xy2
5xy2(x2+4)(x+2)(x2)
48x5y2243xy2
15pq15p+12q12
3(5p+4)(q1)
12ab6a+10b5
4x2+40x+84
4(x+3)(x+7)
5q215q90
4u5v+4u2v3
u2(u+1)(u2u+1)
5m4n+320mn4
4c2+20cd+81d2

prime

25x2+35xy+49y2
10m46250
10(m5)(m+5)(m2+25)
3v4768
36x2y+15xy6y
3y(3x+2)(4x1)
60x2y75xy+30y
8x327y3
(2x3y)(4x2+6xy+9y2)
64x3+125y3
y61
(y+1)(y1)(y2y+1)(y2+y+1)
y6+1
9x26xy+y249
(3xy+7)(3xy7)
16x224xy+9y264
(3x+1)26(3x1)+9
(9x212x+4)
(4x5)27(4x5)+12

Writing Exercises

Explain what it mean to factor a polynomial completely.

Answers will vary.

The difference of squares y4625

can be factored as (y225)(y2+25).

But it is not completely factored. What more must be done to completely factor.

Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.

Answers will vary.

Create three factoring problems that would be good test questions to measure your knowledge of factoring. Show the solutions.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 columns, 1 row and a header row. The header row labels each column: I can, confidently, with some help and no, I don’t get it. The first column has the following statement: recognize and use the appropriate method to factor a polynomial completely. The remaining columns are blank. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


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