Factoring Polynomial Functions pt 2 I Factoring Methods

Factoring Polynomial Functions (pt 2) I. . Factoring Methods. F) Sum & Difference of Cubes (2 terms). a 3 + b 3 = 0 1) Can you 3√ a & 3√ b ? If so, then use. . . a) Sum of Cubes: a 3 + b 3 = (a + b)(a 2 – ab + b 2) b) Diff. of Cubes: a 3 – b 3 = (a – b)(a 2 + ab + b 2) Examples: 8 x 3 + 27 = 0 3√ 8 x 3 = 2 x ; 3√ 27 = 3 64 x 9 – 125 y 3 = 0 3√ 64 x 9 = 4 x 3 ; 3√ 125 y 3 = 5 y (2 x + 3)([2 x]2 – [2 x][3] + 32) (4 x 3 – 5 y)([4 x 3]2 + [4 x 3][5 y] + [5 y]2) (2 x + 3)(4 x 2 – 6 x + 9) (4 x 3 – 5 y)(16 x 6 + 20 x 3 y + 25 y 2)

Factoring Polynomial Functions (pt 2) I. . Factoring Methods. G) Factor by Grouping (4 terms). Uses GCF twice. 1) Group two terms together & factor out the GCF. 2) Group the other two terms & factor out their GCF. 3) Gives GCF 1 (ax + b) + GCF 2 (ax + b). a) The part inside the ( ) will [must] be the same. (ax + b) 4) Write in (GCF 1 + GCF 2) (ax + b) form. Example: 6 x 3 – 15 x 2 + 8 x – 20 = 0 (6 x 3 – 15 x 2) + (8 x – 20) GCF 1 = 3 x 2 ; GCF 2 = 4 3 x 2(2 x – 5) + 4(2 x – 5) notice the inside is the same? (3 x 2 + 4) (2 x – 5)

Factoring Polynomial Functions (pt 2) II. . How many factors / solutions / x-intercepts / roots are there? A) There as many solutions as the degree (exp) of the poly. B) Some solutions (& factors) are repeated. (same x value) C) A function can still be factored as long as one (or more) term still has an exponent on the variable. D) A polynomial function is considered fully factored if… 1) Each term has an integer coefficient (except GCF). 2) No variables have exponents (unless it is the GCF) – or – a) the part with the exponent cannot be factored into terms having integer coefficients.

Factoring Polynomial Functions (pt 2) Fully factored examples: 4 x 3(x + 3)(5 x – 4) has no exp (except the GCF) ½x 7(3 x – 5)(9 x 2 + 11) the (9 x 2 + 11) won’t factor Non-fully factored examples: 4 x 3(x + 3)(4 x 2 – 9) the (4 x 2 – 9) is Difference of squares factors to 4 x 3(x + 3)(2 x – 3) ½x 8(x 2 – 6 x + 8) the (x 2 – 6 x + 8) is Reverse FOIL factors to ½x 8(x – 2)(x – 4)