Trying to help my son out with a homework problem and not sure where I'm going awry. The question is What must be added to the polynomial f(x)= x^5 + x^4 + 3x^3 - 6x^2 - 4x + 8 so that the resulting polynomial is exactly divisible by g(x) = x - 2

The remainder theorem says that if f(x) is divided by g(x)=(x-a) then the remainder is f(a), correct? So in this case, a=2 and we want the remainder to be 0, correct? Thus we set f(2)+Z = 0 and solve for Z.

That leads me to 2^5 + 2^4 + (3)(2)^3 - (6)(2)^2 - (4)(2) + 8 + Z = 0

Which yields Z = -48.

However, this is a multiple choice question, and the choices are

  • -10
  • -18
  • 10
  • 18
  • -8

Where have I gone wrong?

  • $\begingroup$ Crap...missed a term when I was typing it in. Edited the OP...I had that term in my calculations though. $\endgroup$
    – jerH
    Sep 7 '16 at 3:46

You solved this correctly and none of the answer choices work. (As a check, you can use Wolfram Alpha to check the factorization of polynomials. Type "factor" followed by the polynomial in the search bar.)

  • 2
    $\begingroup$ Thanks! I was beginning to feel very inadequate... $\endgroup$
    – jerH
    Sep 7 '16 at 3:53

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