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Factoring Polynomials. Chapter 8.1 Objective 1. Recall: Prime Factorization. Finding the G reatest C ommon F actor of numbers. The GCF is the largest number that will divide into the elements equally. Find the GCF of 3 and 15. 1 st find the prime factors of 3 and 15

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factoring polynomials

Factoring Polynomials

Chapter 8.1 Objective 1

recall prime factorization
Recall: Prime Factorization
  • Finding the Greatest Common Factor of numbers.
  • The GCF is the largest number that will divide into the elements equally.
  • Find the GCF of 3 and 15.
  • 1st find the prime factors of 3 and 15
  • 3=13 15=13 5
  • Determine the GCF by taking common factor (as it occurs the least & occurs in all elements) .
  • 1and 3 occurs in both 3 and 15 so,
  • GCF = 1 3 = 3 (1 can be the GCF of some elements).
find the gcf of variables
Find the GCF of Variables.
  • The GCF is the common variable that will divide into the monomials equally.
  • Find the GCF of x3 and x5.
  • 1st find the prime factors of x3 and x5
  • x3=x x x x5=x x x x x
  • Determine the GCF by largest common factor (as it occurs the least & occurs in all monomials) .
  • x x x occurs in both x3 and x5 so,
  • GCF = x x x = x3
find the gcf of 12a 4 b and 18a 2 b 2 c
Find the GCF of 12a4b and 18a2b2c
  • Find Prime Factors each monomial
  • 12a4b = 2 2 3a a a ab
  • 18a2b2c = 23 3 a ab bc
  • To find GCF consider common factors (must occur in all monomials).
  • GCF = 2 3a2b = 6a2b* c is not in GCF because it does not occur in each monomial*
find the gcf of 4x 6 y and 18x 2 y 6
Find the GCF of 4x6y and 18x2y6
  • Factor each monomial
  • 4x6y = 2 2 x x x x x x y
  • 18x2y6 = 2 3 3 x xy y y y y y
  • To find GCF consider common factors (must occur in all monomials).
  • GCF = 2x2y = 2x2y
factor a polynomial by gcf
Factor a Polynomial by GCF
  • Recall Distributive Property.
  • 5x(x+1) = 5x2 + 5x
  • The objective of factoring out GCF is to extract common factors.
  • Factor 5x2 + 5x by finding GCF.
  • What is the GCF of 5x2 + 5x?
  • 5x is the GCF, but when you factor 5x out, you must divide the polynomial by the GCF. 5x (5x2 + 5x)
  • 5x 5x = 5x(x+1)
factor 14a 2 21a 4 b
Factor 14a2 – 21a4b
  • Find GCF of each monomial.
  • 14a2 = 2 7a a
  • 21a4b = 3 7a a aab GCF = 7a2
  • Factor out GCF
  • 7a2 (14a2 – 21a4b)
  • Divide by GCF7a2 7a2
  • 7a2 (2 - 3a2 b)
factor 6x 4 y 2 9x 3 y 2 12x 2 y 4
Factor. 6x4y2 – 9x3y2 +12x2y4
  • Find GCF of each monomial
  • 6x4y2 = 2 3 x x x xyy
  • 9x3y2 = 3 3 x x xy y
  • 12x2y4 = 2 2 3x xyy y y
  • Factor 3x2y2(6x4y2 – 9x3y2 +12x2y4)
  • Divide by GCF3x2y2 3x2y2 3x2y2
  • 3x2y2 (2x2 – 3x + 4y2)
now you try
NOW YOU TRY!
  • Factor the following.
  • 1. 10y2 – 15y3z
  • 5y2(2 – 3yz)
  • 2. 12m2 +6m -18
  • 6(2m2 + m- 3)
  • 3. 20x4y3 – 30x3y4 +40x2y5
  • 10x2y3 (2x2 - 3xy + 4y2)
  • 4. 13x5y4 – 9x3y2 +12x2y4
  • x2y2 (13x3y2 - 9x + 12y2)
chapter 8 1 objective 2

Chapter 8.1Objective 2

Factor by grouping

when a polynomial has four unlike terms then consider factor by grouping
When a polynomial has four unlike terms, then consider factor by grouping.
  • For the next few examples, the binomials in parenthesis are called binomial factors
  • Factor binomial factors as you would monomials.
  • Factor y(x+2)+3(x+2)
  • (x+2)[y(x+2)+3(x+2)]
  • Divide by GCF (x+2) (x+2)
  • (x+2)[y+3]
  • = (x+2)(y+3)
factor a b 7 b b 7
Factor a(b-7)+b(b-7)
  • Factor binomial factor as you would monomials.
  • (b-7)[a(b-7) +b(b-7)]
  • Divide by GCF (b-7) (b-7)
  • (b-7)[a+b]
  • = (b-7)(a+b)
factor a a b 5 b a
Factor a(a-b)+5(b-a)
  • Notice the binomials are the same except for the signs. You can factor out a -1 from either binomial to make binomials the same
  • a(a-b)+5(-1)(-b+a)
  • Binomials are the same
  • Factor GCF (a-b)[a(a-b)-5(-b+a)]
  • Divide by GCF (a-b) (-b+a)
  • (a-b)[a-5]
  • (a-b) (a-5)
factor 3x 5x 2 4 2 5x
Factor 3x(5x-2) - 4(2-5x)
  • Factor out a -1 from either factor.
  • 3x(-1)(-5x+2)-4(2-5x)
  • -3x(-5x+2)-4(2-5x)
  • Factor GCF (2-5x)[-3x(-5x+2)-4(2-5x)]
  • Divide byGCF(-5x+2) (2-5x)
  • (2-5x) [-3x-4]
  • (2-5x) (-3x- 4)
factor 3y 3 4y 2 6y 8
Factor 3y3-4y2-6y+8
  • Try grouping into binomials to find a binomial factor (sometimes monomials must be rearranged to get binomial factors).
  • GCFy2(3y3- 4y2) GCF-2(-6y+8)
  • y2(3y- 4) -2(3y-4)
  • Factor (3y-4)[y2(3y-4)-2(3y-4)]
  • Divide byGCF(3y-4) (3y-4)
  • (3y-4) [y2 -2]
  • (3y-4) (y2 -2)
factor y 5 5y 3 4y 2 20 by grouping
Factor y5-5y3+4y2-20 by grouping.
  • Find GCFy3(y5-5y3) +4(4y2-20)
  • Divide by GCF y3 y3 4 4
  • y3 (y2-5) +4 (y2-5)
  • Factor Binomial Factor
  • (y2-5)[ y3 (y2-5) +4 (y2-5)]
  • Divide by GCF(y2-5) (y2-5)
  • (y2-5)[y3+4 ]
  • (y2-5)(y3+4 )
now you try1
Now You Try!
  • 1. 6x (4x+3) -5 (4x+3)
  • (4x+3)(6x-5)
  • 2. 8x2- 12x - 6xy + 9y
  • (2x-3)(4x-3y)
  • 3. 7xy2- 3y + 14xy - 6
  • (7xy-3)(y+2)
  • 4. 5xy - 9y – 18 + 10x
  • (5x-9)(y+2)
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