Factoring Polynomials

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# Factoring Polynomials - PowerPoint PPT Presentation

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

Chapter 8.1 Objective 1

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.
• 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 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 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
• 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 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. 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!
• 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 8.1Objective 2

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 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)
• 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 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 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 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 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)