factoring polynomials chapter 8.1 objective 1. recall: prime factorization finding the greatest...

Factoring PolynomialsChapter 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=1 3 15=1 3 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 3 a a a a b

•18a2b2c = 2 3 3 a a b b c

•To find GCF consider common factors (must occur in all monomials).

•GCF = 2 3 a2 b = 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 x y y y y y y

•To find GCF consider common factors (must occur in all monomials).

•GCF = 2 x2 y = 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 7 a a• 21a4b = 3 7 a a a a b GCF = 7a2

•Factor out GCF• 7a2 (14a2 – 21a4b)•Divide by GCF 7a2 7a2

• 7a2 (2 - 3a2 b)

Factor. 6x4y2 – 9x3y2 +12x2y4

•Find GCF of each monomial• 6x4y2 = 2 3 x x x x y y• 9x3y2 = 3 3 x x x y y•12x2y4 = 2 2 3 x x y y y y

•Factor 3x2y2 (6x4y2 – 9x3y2 +12x2y4)•Divide by GCF 3x2y2 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 8.1 Objective 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 by GCF (-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).

• GCF y2(3y3- 4y2) GCF -2(-6y+8)• y2(3y- 4) -2(3y-4)• • Factor (3y-4)[y2(3y-4)-2(3y-4)]• Divide by GCF (3y-4) (3y-4)• (3y-4) [y2 -2]

• (3y-4) (y2 -2)

Factor y5-5y3+4y2-20 by grouping.• Find GCF y3(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)