Factoring Polynomial Functions (pt 2) I.. Factoring Methods. F) Sum & Difference of Cubes (2 terms). a3 + b3 = 0 1) Can you 3√ a & 3√ b ? If so, then use ... a) Sum of Cubes: a3 + b3 = (a + b)(a2 – ab + b2) b) Diff. of Cubes: a3 – b3 = (a – b)(a2 + ab + b2) Examples: 8x3 + 27 = 0 64x9 – 125y3 = 0 3√ 8x3 = 2x ; 3√ 27 = 3 3√ 64x9 = 4x3 ; 3√ 125y3 = 5y (2x + 3)([2x]2 – [2x][3] + 32) (4x3 – 5y)([4x3]2 + [4x3][5y] + [5y]2) (2x + 3)(4x2 – 6x + 9) (4x3 – 5y)(16x6 + 20x3y + 25y2)

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 GCF1 (ax + b) + GCF2 (ax + b). a) The part inside the ( ) will [must] be the same. (ax + b) 4) Write in (GCF1 + GCF2) (ax + b) form. Example: 6x3 – 15x2 + 8x – 20 = 0 (6x3 – 15x2) + (8x – 20) GCF1 = 3x2 ; GCF2 = 4 3x2(2x – 5) + 4(2x – 5) notice the inside is the same? (3x2 + 4) (2x – 5)

Factoring Polynomial Functions (pt 2) II.. How many factors / solutions / x-intercepts / roots are there? A) There are 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: 4x3(x + 3)(5x – 4) has no exp (except the GCF) ½x7(3x – 5)(9x2 + 11) the (9x2 + 11) won’t factor Non-fully factored examples: 4x3(x + 3)(4x2 – 9) the (4x2 – 9) is Difference of squares factors to 4x3(x + 3)(2x + 3)(2x – 3) ½x8(x2 – 6x + 8) the (x2 – 6x + 8) is Reverse FOIL factors to ½x8(x – 2)(x – 4)