Sec3 AMaths - Polynomials

• Only-Way-4-Destiny!

  7 Apr 09, 19:05

  Hi all, I need some help on this question. 

   

  Given that 2x² + 3px - 2q and x² + q have a common factor x - a, where p, q and a are non-zero constants, show that 9p² + 16q = 0.

   

  Thanks.

• dkcx

  7 Apr 09, 20:08

  Replace x by a in both equations since (x-a) means x=a.

  After that, from the 2nd equation, you can get a in terms of q, sub this a value into the 1st equation, do some manipulation and you will get the answer.

• wee_ws

  7 Apr 09, 20:42

  Hi,

    Let f(x) = 2x² + 3px - 2q and g(x) = x² + q.

    Then f(a) = 0 and g(a) = 0, by factor theorem.

    So 2a² + 3pa - 2q = 0 -- (1)

    and a² + q = 0 -- (2).

    From (2), a² = -q, which we substitute into (1) to obtain:

    -2q + 3pa - 2q = 0

    => 3pa = 4q

    => a = 4q/3p (remember that p is non-zero, and q is non-zero)

    Substitute a into (2), we obtain: 

    (4q/3p)² = -q

    => 16q² = -9p²q

    => q(16q + 9p²) = 0, but q is non-zero, so ...

    Thanks!

  Cheers,
  Wen Shih

• Only-Way-4-Destiny!

  7 Apr 09, 21:07

  Thanks!

• SBS n SMRT

  8 Apr 09, 23:21

  thanks, i also blur of this qns