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Chapters 8 and 9 Greatest Common Factors

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Chapters 8 and 9 Greatest Common Factors & Factoring by Grouping Definitions Factor, Factoring, Prime Polynomial Common Factor of 2 or more terms – PowerPoint PPT presentation

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Title: Chapters 8 and 9 Greatest Common Factors


1
Chapters 8 and 9 Greatest Common Factors
Factoring by Grouping
  • Definitions
  • Factor, Factoring, Prime Polynomial
  • Common Factor of 2 or more terms
  • Factoring a Monomial into two factors
  • Identifying Common Monomial Factors
  • Factoring Out Common Factors
  • Arranging a 4 Term Polynomial into Groups
  • Factoring Out Common Binomials

2
Whats a Polynomial Factor?
product (factor)(factor)(factor) (factor)
Factoring is the reverse of multiplication. 84 is
a product that can be expressed by many different
factorizations 84 2(42) or 84 7(12) or
84 4(7)(3) or 84 2(2)(3)(7) Only one
example, 84 2(2)(3)(7), shows 84 as the product
of prime integers. Always try to factor a
polynomial into prime polynomials
3
Factoring Monomials
  • 12x3 also can be expressed in many ways 12x3
    12(x3) 12x3 4x2(3x) 12x3 2x(6x2)
  • Usually, we only look for two factors You try
  • 4a
  • 2(2a) or 4(a)
  • x3
  • x(x2) or x2(x)
  • 14y2
  • 14(y2) or 14y(y) or 7(2y2) or 7y(2y) or
    y(14y)
  • 43x5
  • 43(x5) or 43x(x4) or x3(43x2) or 43x2(x3)
    or

4
Common Factors of Polynomials
  • When a polynomial has 2 or more terms, it may
    have common factors
  • By definition, a common factor must divide
    evenly into every term
  • For x2 3x the only common factor is x , so
  • x2 3x xx x3 x (? ?) x(x 3)
  • For 8y2 12y 20 a common factor is 2, so
  • 8y2 12y 20 2(? ? ?) 2(4y2 6y 10)
  • Check factoring by multiplying
  • 2(4y2 6y 10) 8y2 12y 20

5
The Greatest Common Factor of Polynomials
  • The greatest common factor (or GCF) is the
    largest monomial that can divide evenly into
    every term
  • Looking for common factors in 2 or more terms
    is always the first step in factoring
    polynomials
  • Remember a(b c) ab ac (distributive
    law)
  • Consider that a is a common factor of ab ac
  • If we find a polynomial has form ab ac we can
    factor it into a(b c)
  • For 3x2 3x the greatest common factor is 3x
    , so
  • 3x2 3x 3xx 3x1 3x (? ?) 3x(x 1)
  • Another example 8y2 12y 20
  • The GCF is 4 Divide each term by 4
  • 8y2 12y 20 4(? ? ?) 4(2y2 3y 5)
  • Check by multiplying 4(2y2) 4(3y) 4(5)
    8y2 12y 20

6
Practice Find the Greatest Common Monomial
Factor
  • 7a 21
  • 7(? ?)
  • 7(a 3)
  • 19y3 3y
  • y(? ?)
  • y(19y2 3)
  • 8x2 14x 4
  • 2(? ? ?)
  • 2(4x2 7x 2)
  • 4y2 6y
  • 2y(? ?)
  • 2y(2y 3)

7
Find the Greatest Common Factor
  • 18y5 12y4 6y3
  • 6y3(? ? ?)
  • 6y3(3y2 2y 1)
  • 21x2 42xy 28y2
  • 7(? ? ?)
  • 7(3x2 6xy 4y2)
  • 22x3 110xy2
  • 22x(? ?)
  • 22x(x2 5y2)
  • 7x2 11xy 13y2
  • No common factor exists

8
Introduction to Factoring by GroupingFactoring
Out Binomials
  • x2(x 7) 3(x 7)
  • (x 7)(? ?)
  • (x 7)(x2 3)
  • y3(a b) 2(a b)
  • (a b)(? ?)
  • (a b)(y3 2)

9
PracticeFactoring Out Binomials
  • You try 2x2(x 1) 6x(x 1) 17(x 1)
  • (x 1)(? ? ?)
  • (x 1)(2x2 6x 17)
  • y2(2y 5) x2(2y 5)
  • (2y 5)(? ?)
  • (2y 5)(y2 x2)
  • 5x2(xy 1) 6y(xy 1)
  • No common factors

10
Factoring by Grouping
  • For polynomials with 4 terms
  • Arrange the terms in the polynomial into 2 groups
  • Factor out the common monomials from each group
  • If the binomial factors produced are either
    identical or opposite, complete the factorization
  • Example 2c 2d cd d2
  • 2(c d) d(c d)
  • (c d)(2 d)

11
Factor by Grouping
  • 8t3 2t2 12t 3
  • 2t2(4t 1) 3(4t 1)
  • (4t 1)(2t2 3)

12
Factor by Grouping
  • 4x3 6x2 6x 9
  • 2x2(2x 3) 3(2x 3)
  • (2x 3)(2x2 3)

13
Factor by Grouping
  • y4 2y3 12y 3
  • y3(y 2) 3(4y 1)
  • Oops not factorable via grouping

14
Grouping Unusual Polynomials
  • x3 7x2 6x x2y 7xy 6y
  • x(x2 7x 6) y(x2 7x 6)
  • (x2 7x 6)(x y)
  • (x 1)(x 6)(x y)

15
What Next?
  • Section 5.6 Factoring Trinomials
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