TTMUA Lab
TMUA 2016 · Paper 1 · Question 2 of 20

TMUA 2016 Paper 1 Question 2

Algebra and functions — Factor theorem · complete factorisation of a cubic. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1Algebra and functionsFactor theorem · complete factorisation of a cubic6 options
The expression 3x3 + 13x2 + 8x + a, where a is a constant, has (x + 2) as a factor.

Which one of the following is a complete factorisation of the expression?

  1. A(x+2)(x−1)(3x−2)
  2. B(x+2)(x+1)(3x−2)
  3. C(x+2)(x+1)(3x+2)
  4. D(x+2)(x−3)(3x+2)
  5. E(x+2)(x+3)(3x−2)
  6. F(x+2)(x+3)(3x+2)
Show the answer and worked solution
answer · E
  1. A(x+2)(x−1)(3x−2)
  2. B(x+2)(x+1)(3x−2)
  3. C(x+2)(x+1)(3x+2)
  4. D(x+2)(x−3)(3x+2)
  5. E(x+2)(x+3)(3x−2)
  6. F(x+2)(x+3)(3x+2)
The factor theorem gives f(−2) = 0: 3(−8) + 13(4) + 8(−2) + a = −24 + 52 − 16 + a = 0, so a = −12. Dividing 3x3 + 13x2 + 8x − 12 by x + 2 leaves 3x2 + 7x − 6, which factorises as (3x − 2)(x + 3). Hence the complete factorisation is (x+2)(x+3)(3x−2). You can also settle it without division by checking which option has constant term −12 and x2 coefficient matching 13.