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Factoring Polynomials

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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  1. Factoring Polynomials Chapter 8.1 Objective 1

  2. 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).

  3. 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

  4. 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*

  5. 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

  6. 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)

  7. 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)

  8. 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)

  9. 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)

  10. Chapter 8.1Objective 2 Factor by grouping

  11. 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)

  12. 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)

  13. 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)

  14. 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)

  15. 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)

  16. 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 )

  17. 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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