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### Factoring Polynomials

### Chapter 8.1Objective 2

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)

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)

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