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TMUA 2020 · Paper 1 · Question 2 of 20

TMUA 2020 Paper 1 Question 2

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

TMUA 2020 · Paper 1Algebra and functionsFactor theorem · two factors of a cubic6 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
Note there is no x term, so the cubic is 2x3 + px2 + 0x + q. The factor theorem gives two equations: x = 2 yields 16 + 4p + q = 0, and x = −12 yields −14 + p4 + q = 0, that is −1 + p + 4q = 0. From the first, q = −16 − 4p; substituting gives −1 + p − 64 − 16p = 0, so p = −133 and q = 43. Hence 2p + q = −263 + 43 = −223.