How to find factor when the polynomial is not given

For the question " If $f(x)$ is a polynomial with constant term $10$ having a factor $(x-k)$ where $k$ is an integer, then find the possible value of $k$", the options given are $-20, 20, 8$ and $5$.

I get that by remainder theorem, $k$ should divide $f(x)$, so, $f(k)$ would be zero.

Since $f(x)$ is not given, I assumed that to be $[x/2 + 10]$ to satisfy the first option. Same way, other polynomials can also be imagined and all options would satisfy. But the answer is given as only $5$.

If they had asked least positive value of $k$ then I may have understood but that is not the case. They have clearly asked 'possible value of $k$', so all the options should be correct.

But as is usually the case, I miss some fine detail, so, need your help yo figure that out. Thanks.

You're correct: for the problem as stated, $\,k\,$ can be any integer, e.g. $\ 10/k\ (x-k) = 10x/k - 10.\$

Probably intended was that the polynomial has integer coefficients. Then the result follows by the Rational Root Test, or Gauss's Lemma, or the division algorithm.

• I see, thats very interesting! If the leading coefficient is a fraction, all the given options are a possibility xD (every integer is a possibile root). Question does need some revision! thanks for pointing it out :) Sep 5, 2014 at 14:52
• @ganeshie8 Yes. Note, in particular, that the Rational Root Test applies only to polynomials with integer coefficients, so it cannot be applied to the problem as currently stated. Sep 5, 2014 at 15:01

HINT

rational root test

What are the factors of constant term, $10$ ?

• 1, 2, 5, 10. So either of these coud have been the 'possible zero', not really the actual zero, because of this test. That's interesting. Thanks. Sep 5, 2014 at 12:04
• Exactly ! and notice that we don't need to know the actual leading coefficient to find the possible integer roots; constant term is sufficient Sep 5, 2014 at 12:05
• @ganeshie8 This is not true without further hypotheses - see my answer. Sep 5, 2014 at 14:45