Factoring polynomials
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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

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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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