Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let P(x) be a real polynomial of degree less than or equal to 4, such that there are at least 5 distinct solutions to P(x) = 5. Find P(5).

I've been trying to wrap my head around this supposedly basic problem, but I keep running into a blank wall. I can get at most 4 distinct solutions to P(x) = 5 for a given degree 4 polynomial, but I keep coming to the conclusion that 5 distinct solutions is impossible. For example, I used P(x) = x^4 - 2x^2 + 5.5. Given this P(x) it is indeed possible to find 4 distinct x's for which P(x) = 5. What am I missing here? Is there perhaps a complex solution, and if so, how would I find it?

share|cite|improve this question
up vote 2 down vote accepted

Hint: What if there were much more than five distinct solutions to $P(x)=5$? Can you think of how this could be possible? Try to think of lower degree polynomials, rather than degree four.

share|cite|improve this answer
Wow, Ok, I think I might have been the world's biggest idiot. Correct me if I am wrong, but a polynomial of degree zero is still a polynomial. If that statement is true, then all that remains to be done is allow P(x) = 5, and thus all the requirements on P are met, and I can confidently say that P(5) = 5. – user79790 Aug 12 '13 at 23:57
@user79790 That's pretty much it. The key point is a non-zero $n$-degree polynomial can have at most $n$ roots, but $P(x)-5$ has $5$ roots. Since $P(x)-5$ is of degree less than or equal to $4$ it must be zero. – JSchlather Aug 13 '13 at 0:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.