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 st 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 x 3 and x 5. 1 st find the prime factors of x 3 and x 5 x 3 =x x x x 5 =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 x 3 and x 5 so, GCF = x x x = x 3

Find the GCF of 12a 4 b and 18a 2 b 2 c Find Prime Factors each monomial 12a 4 b = a a a a b 18a 2 b 2 c = a a b b c To find GCF consider common factors (must occur in all monomials). GCF = 2 3 a 2 b = 6a 2 b * 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 Factor each monomial 4x 6 y = 2 2 x x x x x x y 18x 2 y 6 = x x y y y y y y To find GCF consider common factors (must occur in all monomials). GCF = 2 x 2 y = 2x 2 y

Factor a Polynomial by GCF Recall Distributive Property. 5x(x+1) = 5x 2 + 5x The objective of factoring out GCF is to extract common factors. Factor 5x 2 + 5x by finding GCF. What is the GCF of 5x 2 + 5x? 5x is the GCF, but when you factor 5x out, you must divide the polynomial by the GCF. 5x (5x 2 + 5x) 5x 5x = 5x(x+1)

Factor 14a 2 – 21a 4 b Find GCF of each monomial. 14a 2 = 2 7 a a 21a 4 b = 3 7 a a a a b GCF = 7a 2 Factor out GCF 7a 2 (14a 2 – 21a 4 b) Divide by GCF 7a 2 7a 2 7a 2 (2 - 3a 2 b)

Factor. 6x 4 y 2 – 9x 3 y 2 +12x 2 y 4 Find GCF of each monomial 6x 4 y 2 = 2 3 x x x x y y 9x 3 y 2 = 3 3 x x x y y 12x 2 y 4 = x x y y y y Factor 3x 2 y 2 (6x 4 y 2 – 9x 3 y 2 +12x 2 y 4 ) Divide by GCF 3x 2 y 2 3x 2 y 2 3x 2 y 2 3x 2 y 2 (2x 2 – 3x + 4y 2 )

NOW YOU TRY! Factor the following y 2 – 15y 3 z 5y 2 (2 – 3yz) 2. 12m 2 +6m -18 6(2m 2 + m- 3) 3. 20x 4 y 3 – 30x 3 y 4 +40x 2 y 5 10x 2 y 3 (2x 2 - 3xy + 4y 2 ) 4. 13x 5 y 4 – 9x 3 y 2 +12x 2 y 4 x 2 y 2 (13x 3 y 2 - 9x + 12y 2 )

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 3y 3 -4y 2 -6y+8 Try grouping into binomials to find a binomial factor (sometimes monomials must be rearranged to get binomial factors). GCF y 2 (3y 3 - 4y 2 ) GCF -2(-6y+8) y 2 (3y- 4) -2(3y-4) Factor (3y-4)[y 2 (3y-4)-2(3y-4)] Divide by GCF (3y-4) (3y-4) (3y-4) [y 2 -2] (3y-4) (y 2 -2)

Factor y 5 -5y 3 +4y by grouping. Find GCF y 3 (y 5 -5y 3 ) +4(4y 2 -20) Divide by GCF y 3 y y 3 (y 2 -5) +4 (y 2 -5) Factor Binomial Factor (y 2 -5)[ y 3 (y 2 -5) +4 (y 2 -5)] Divide by GCF (y 2 -5) (y 2 -5) (y 2 -5)[y 3 +4 ] (y 2 -5)(y 3 +4 )

Now You Try! 1. 6x (4x+3) -5 (4x+3) (4x+3)(6x-5) 2. 8x x - 6xy + 9y (2x-3)(4x-3y) 3. 7xy 2 - 3y + 14xy - 6 (7xy-3)(y+2) 4. 5xy - 9y – x (5x-9)(y+2)