1. ExampleFactorisex 3 + 3x 2 – 10x - 24 We need some trial & error with factors of –24 ie +/-1, +/-2, +/-3 etc f(-1) = -1 13-10-24 1 2 -2 -12 12 -12No good f(1) = 113-10-24 1 1 4 4 -6 -30No good

2. f(-2) = -213-10-24 1 -2 1 -12 24 0 f(-2) = 0 so (x + 2) a factor Other factor is x 2 + x - 12= (x + 4)(x – 3) So x 3 + 3x 2 – 10x – 24 = (x + 4)(x + 2)(x – 3) Roots/Zeros The roots or zeros of a polynomial tell us where it cuts the X-axis. ie where f(x) = 0. If a cubic polynomial has zeros a, b & c then it has factors (x – a), (x – b) and (x – c).

3. ExampleSolvex 4 + 2x 3 - 8x 2 – 18x – 9 = 0 We need some trial & error with factors of –9 ie +/-1, +/-3 etc f(-1) = -1 12-8-18-9 1 1 -9 9 9 0 f(-1) = 0 so (x + 1) a factor Other factor is x 3 + x 2 – 9x - 9 which we can call g(x) test +/-1, +/-3 etc

4. g(-1) = -111-9-9 1 0 0 -9 9 0 g(-1) = 0 so (x + 1) a factor Other factor is x 2 – 9= (x + 3)(x – 3) if x 4 + 2x 3 - 8x 2 – 18x – 9 = 0 then (x + 3)(x + 1)(x + 1)(x – 3) = 0 So x = -3 or x = -1 or x = 3

5. Breakdowns A cubic polynomialie ax 3 + bx 2 + cx + d could be factorised into either (i) Three linear factors of the form (x + a) or (ax + b) or (ii) A linear factor of the form (x + a) or (ax + b) and a. quadratic factor (ax 2 + bx + c) which doesn’t factorise. or (iii) It may be irreducible. IT DIZNAE FACTORISE