f(x) = 3x^3 – x^2 – 20x – 12 (a) Use the factor theorem to show that (3x + 2) is a factor of f(x). [2 marks] (b) Factorise f(x) fully. [3 marks]
(a) Factor theorem hence, use x = -2/3. Sub in : 3(-2/3)3 - (-2/3)2 -20 (-2/3) -12 = 0 (b) (3x+2)(ax2 + bx+c) = 3x3 – x2 – 20x – 12 3ax3 + (3b+2a)x2 + (3c+2b)x + 2c = 3x3 – x2 – 20x – 12 Therefore, 3a = 3 ----------> a = 1 3b +2a = -1 -----> b = -1 2c = -12 ---------> c = -6factorise: x2 - x- 6 = (x+2)(x-3) Fully factorised: 3x3 – x2 – 20x – 12 = (3x+2)(x+2)(x-3)
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