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6.5: General Strategy for Factoring Polynomial Expressions

  • Page ID
    114194
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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.

    How To

    Use a general strategy for factoring polynomials.

    1. Step 1. Is there a greatest common factor?
      Factor it out.
    2. Step 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?x2+bx+c? Undo FOIL.
      • Is it of the form ax2+bx+c?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. Step 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 6.35

    Factor completely: 7x321x270x.7x321x270x.

    Answer
      7x321x270x7x321x270x
    Is there a GCF? Yes, 7x7x.  
    Factor out the GCF. 7x(x23x10)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)(x)
      7x(x+2)(x5)7x(x+2)(x5)
    Is the expression factored completely? Yes.  
    Neither binomial can be factored.  
    Check your answer.  
    Multiply.  
    7x(x+2)(x5)7x(x+2)(x5)  
    7x(x25x+2x10)7x(x25x+2x10)  
    7x(x23x10)7x(x23x10)  
    7x321x270x7x321x270x  
    Try It 6.69

    Factor completely: 8y3+16y224y.8y3+16y224y.

    Try It 6.70

    Factor completely: 5y315y2270y.5y315y2270y.

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

    Example 6.36

    Factor completely: 24y2150.24y2150.

    Answer
      24y215024y2150
    Is there a GCF? Yes, 6.  
    Factor out the GCF. 6(4y225)6(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. 6((2y)2(5)2)6((2y)2(5)2)
    Write as a product of conjugates. 6(2y5)(2y+5)6(2y5)(2y+5)
    Is the expression factored completely?Is the expression factored completely?  
    Neither binomial can be factored.Neither binomial can be factored.  
    Check:  
    Multiply.Multiply.  
    6(2y5)(2y+5)6(2y5)(2y+5)  
    6(4y225)6(4y225)  
    24y215024y2150  
    Try It 6.71

    Factor completely: 16x336x.16x336x.

    Try It 6.72

    Factor completely: 27y248.27y248.

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

    Example 6.37

    Factor completely: 4a212ab+9b2.4a212ab+9b2.

    Answer
      4a212ab+9b24a212ab+9b2
    Is there a GCF? No.  
    Is it a binomial, trinomial, or are there more terms?  
    Trinomial with a1a1. But the first term is a perfect square.  
    Is the last term a perfect square? Yes. (2a)212ab+(3b)2(2a)212ab+(3b)2
    Does it fit the pattern, a22ab+b2a22ab+b2? Yes. (2a)2−12ab+−2(2a)(3b)(3b)2(2a)2−12ab+−2(2a)(3b)(3b)2
    Write it as a square. (2a3b)2(2a3b)2
    Is the expression factored completely? Yes.Is the expression factored completely? Yes.  
    The binomial cannot be factored.The binomial cannot be factored.  
    Check your answer.  
    Multiply.Multiply.  
    (2a3b)2(2a3b)2  
    (2a)22·2a·3b+(3b)2(2a)22·2a·3b+(3b)2  
    4a212ab+9b24a212ab+9b2
    Try It 6.73

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

    Try It 6.74

    Factor completely: 9x224xy+16y2.9x224xy+16y2.

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

    Example 6.38

    Factor completely 12x3y2+75xy2.12x3y2+75xy2.

    Answer
      12x3y2+75xy212x3y2+75xy2
    Is there a GCF? Yes, 3xy23xy2.  
    Factor out the GCF. 3xy2(4x2+25)3xy2(4x2+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.
    Is the expression factored completely? Yes.Is the expression factored completely? Yes.  
    Check:  
    Multiply.Multiply.  
    3xy2(4x2+25)3xy2(4x2+25)  
    12x3y2+75xy212x3y2+75xy2
    Try It 6.75

    Factor completely: 50x3y+72xy.50x3y+72xy.

    Try It 6.76

    Factor completely: 27xy3+48xy.27xy3+48xy.

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

    Example 6.39

    Factor completely: 24x3+81y3.24x3+81y3.

    Answer
    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 6.77

    Factor completely: 250m3+432n3.250m3+432n3.

    Try It 6.78

    Factor completely: 2p3+54q3.2p3+54q3.

    Example 6.40

    Factor completely: 3x5y48xy.3x5y48xy.

    Answer
      3x5y48xy3x5y48xy
    Is there a GCF? Factor out 3xy3xy 3xy(x416)3xy(x416)
    Is the binomial a sum or difference? Of squares or cubes?
    Write it as a difference of squares.
    3xy((x2)2(4)2)3xy((x2)2(4)2)
    Factor it as a product of conjugates 3xy(x24)(x2+4)3xy(x24)(x2+4)
    The first binomial is again a difference of squares. 3xy((x)2(2)2)(x2+4)3xy((x)2(2)2)(x2+4)
    Factor it as a product of conjugates. 3xy(x2)(x+2)(x2+4)3xy(x2)(x+2)(x2+4)
    Is the expression factored completely? Yes.  
    Check your answer.  
    Multiply.  
    3xy(x2)(x+2)(x2+4)3xy(x2)(x+2)(x2+4)  
    3xy(x24)(x2+4)3xy(x24)(x2+4)  
    3xy(x416)3xy(x416)  
    3x5y48xy3x5y48xy  
    Try It 6.79

    Factor completely: 4a5b64ab.4a5b64ab.

    Try It 6.80

    Factor completely: 7xy57xy.7xy57xy.

    Example 6.41

    Factor completely: 4x2+8bx4ax8ab.4x2+8bx4ax8ab.

    Answer
      4x2+8bx4ax8ab4x2+8bx4ax8ab
    Is there a GCF? Factor out the GCF, 4. 4(x2+2bxax2ab)4(x2+2bxax2ab)
    There are four terms. Use grouping. 4[x(x+2b)a(x+2b)]4(x+2b)(xa)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+8bx4ax8ab4(x+2b)(xa)4(x2ax+2bx2ab)4x2+8bx4ax8ab
     
    Try It 6.81

    Factor completely: 6x212xc+6bx12bc.6x212xc+6bx12bc.

    Try It 6.82

    Factor completely: 16x2+24xy4x6y.16x2+24xy4x6y.

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

    Example 6.42

    Factor completely: 40x2y+44xy24y.40x2y+44xy24y.

    Answer
      40x2y+44xy24y40x2y+44xy24y
    Is there a GCF? Factor out the GCF, 4y4y. 4y(10x2+11x6)4y(10x2+11x6)
    Factor the trinomial with a1a1. 4y(10x2+11x6)4y(10x2+11x6)
      4y(5x2)(2x+3)4y(5x2)(2x+3)
    Is the expression factored completely? Yes.  
    Check your answer.  
    Multiply.  
    4y(5x2)(2x+3)4y(5x2)(2x+3)  
    4y(10x2+11x6)4y(10x2+11x6)  
    40x2y+44xy24y40x2y+44xy24y  
    Try It 6.83

    Factor completely: 4p2q16pq+12q.4p2q16pq+12q.

    Try It 6.84

    Factor completely: 6pq29pq6p.6pq29pq6p.

    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 6.43

    Factor completely: 9x212xy+4y249.9x212xy+4y249.

    Answer
      9x212xy+4y2499x212xy+4y249
    Is there a GCF? No.  
    With more than 3 terms, use grouping. Last 2 terms have no GCF. Try grouping first 3 terms. 9x212xy+4y2499x212xy+4y249
    Factor the trinomial with a1a1. But the first term is a perfect square.  
    Is the last term of the trinomial a perfect square? Yes. (3x)212xy+(2y)249(3x)212xy+(2y)249
    Does the trinomial fit the pattern, a22ab+b2a22ab+b2? Yes. (3x)2−12xy+−2(3x)(2y)(2y)249(3x)2−12xy+−2(3x)(2y)(2y)249
    Write the trinomial as a square. (3x2y)249(3x2y)249
    Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares. (3x2y)272(3x2y)272
    Write it as a product of conjugates. ((3x2y)7)((3x2y)+7)((3x2y)7)((3x2y)+7)
      (3x2y7)(3x2y+7)(3x2y7)(3x2y+7)
    Is the expression factored completely? Yes.  
    Check your answer.  
    Multiply.  
    (3x2y7)(3x2y+7)(3x2y7)(3x2y+7)  
    9x26xy21x6xy+4y2+14y+21x14y499x26xy21x6xy+4y2+14y+21x14y49  
    9x212xy+4y2499x212xy+4y249  
    Try It 6.85

    Factor completely: 4x212xy+9y225.4x212xy+9y225.

    Try It 6.86

    Factor completely: 16x224xy+9y264.16x224xy+9y264.

    Section 6.4 Exercises

    Practice Makes Perfect

    Recognize and Use the Appropriate Method to Factor a Polynomial Completely

    In the following exercises, factor completely.

    233.

    2 n 2 + 13 n 7 2 n 2 + 13 n 7

    234.

    8 x 2 9 x 3 8 x 2 9 x 3

    235.

    a 5 + 9 a 3 a 5 + 9 a 3

    236.

    75 m 3 + 12 m 75 m 3 + 12 m

    237.

    121 r 2 s 2 121 r 2 s 2

    238.

    49 b 2 36 a 2 49 b 2 36 a 2

    239.

    8 m 2 32 8 m 2 32

    240.

    36 q 2 100 36 q 2 100

    241.

    25 w 2 60 w + 36 25 w 2 60 w + 36

    242.

    49 b 2 112 b + 64 49 b 2 112 b + 64

    243.

    m 2 + 14 m n + 49 n 2 m 2 + 14 m n + 49 n 2

    244.

    64 x 2 + 16 x y + y 2 64 x 2 + 16 x y + y 2

    245.

    7 b 2 + 7 b 42 7 b 2 + 7 b 42

    246.

    30 n 2 + 30 n + 72 30 n 2 + 30 n + 72

    247.

    3 x 4 y 81 x y 3 x 4 y 81 x y

    248.

    4 x 5 y 32 x 2 y 4 x 5 y 32 x 2 y

    249.

    k 4 16 k 4 16

    250.

    m 4 81 m 4 81

    251.

    5 x 5 y 2 80 x y 2 5 x 5 y 2 80 x y 2

    252.

    48 x 5 y 2 243 x y 2 48 x 5 y 2 243 x y 2

    253.

    15 p q 15 p + 12 q 12 15 p q 15 p + 12 q 12

    254.

    12 a b 6 a + 10 b 5 12 a b 6 a + 10 b 5

    255.

    4 x 2 + 40 x + 84 4 x 2 + 40 x + 84

    256.

    5 q 2 15 q 90 5 q 2 15 q 90

    257.

    4 u 5 + 4 u 2 v 3 4 u 5 + 4 u 2 v 3

    258.

    5 m 4 n + 320 m n 4 5 m 4 n + 320 m n 4

    259.

    4 c 2 + 20 c d + 81 d 2 4 c 2 + 20 c d + 81 d 2

    260.

    25 x 2 + 35 x y + 49 y 2 25 x 2 + 35 x y + 49 y 2

    261.

    10 m 4 6250 10 m 4 6250

    262.

    3 v 4 768 3 v 4 768

    263.

    36 x 2 y + 15 x y 6 y 36 x 2 y + 15 x y 6 y

    264.

    60 x 2 y 75 x y + 30 y 60 x 2 y 75 x y + 30 y

    265.

    8 x 3 27 y 3 8 x 3 27 y 3

    266.

    64 x 3 + 125 y 3 64 x 3 + 125 y 3

    267.

    y 6 1 y 6 1

    268.

    y 6 + 1 y 6 + 1

    269.

    9 x 2 6 x y + y 2 49 9 x 2 6 x y + y 2 49

    270.

    16 x 2 24 x y + 9 y 2 64 16 x 2 24 x y + 9 y 2 64

    271.

    ( 3 x + 1 ) 2 6 ( 3 x + 1 ) + 9 ( 3 x + 1 ) 2 6 ( 3 x + 1 ) + 9

    272.

    ( 4 x 5 ) 2 7 ( 4 x 5 ) + 12 ( 4 x 5 ) 2 7 ( 4 x 5 ) + 12

    Writing Exercises

    273.

    Explain what it mean to factor a polynomial completely.

    274.

    The difference of squares y4625y4625 can be factored as (y225)(y2+25).(y225)(y2+25). But it is not completely factored. What more must be done to completely factor.

    275.

    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.

    276.

    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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