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TMUA 2019 · Paper 2 · Question 2 of 20

TMUA 2019 Paper 2 Question 2

Algebra and functions — Factor theorem · a cubic with a missing term. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 2Algebra and functionsFactor theorem · a cubic with a missing term6 options
(2x + 1) and (x − 2) are factors of 2x3 + px2 + q.

What is the value of 2p + q?

  1. A−10
  2. B−385
  3. C−223
  4. D223
  5. E385
  6. F10
Show the answer and worked solution
answer · C
  1. A−10
  2. B−385
  3. C−223
  4. D223
  5. E385
  6. F10
Write f(x) = 2x3 + px2 + 0x + q — the point to notice is that the coefficient of x is zero, so there are only two unknowns. The factor theorem gives f(2) = 16 + 4p + q = 0 and f!(−12) = −14 + p4 + q = 0, the second of which tidies to p + 4q = 1. Substituting p = 1 − 4q into 4p + q = −16 gives −15q = −20, so q = 43 and p = −133. Then 2p + q = −263 + 43 = −223.