Problem related to polynomial which takes the same value for three inputs

Question

Let p(x) be a polynomial over the integers such that it takes 1 value at three different integers. Prove that it has no integral root.

MY ATTEMPT:

I couldn't really think of anything. I first thought that this polynomial cannot be linear or quadratic, as it takes the same values at more than two points(an application of the identity theorem for polynomials) but I dont know where to take that.

This is not clear. If, say, $p(x)=x(x-1)(x+1)$. then $p(x)=0$ at each of $0,1,-1$ and of course it has integer roots. — lulu, Apr 04 '20 at 16:32
I've written it exactly as it's worded in this book I have. There's nothing wrong with your counterexample, so this seems to be a wrong question. I'll delete this — MNIShaurya, Apr 04 '20 at 16:36
Do they really write "1 value" where one should write "one value"? — Hagen von Eitzen, Apr 04 '20 at 16:36
I'm guessing it is $P(a)=P(b)=P(c)=1$, show that $P(x)=0$ does not have an integer $x$ solution. — Calvin Lin, Apr 04 '20 at 16:40
@CalvinLin If they did mean that, it would be the most ambiguous of ambiguous wording. Regardless, I'll try the '1 as a value' version on my own — MNIShaurya, Apr 04 '20 at 16:44

score 3 · Answer 1 · answered Apr 04 '20 at 16:51

Let $q(x)=p(x)-1$

$q(a)=q(b)=q(c)=0$ for some integers $a$, $b$, $c$.

$q(x)=(x-a)(x-b)(x-c)\cdot r(x)$, where $r(x)$ is some other polynomial over the integers.

When $p(x)=0$, $(x-a)(x-b)(x-c)\cdot r(x) = -1$

This is impossible since $a,b,c$ are integers, so the closest $(x-a)(x-b)(x-c)$ can be is $2$ or $-2$

score 2 · Accepted Answer · answered Apr 04 '20 at 16:39

2

(See my comment for my interpretation of the question.)

Hint: For integer polynomials, $a-b \mid P(a) - P(b)$.

answered Apr 04 '20 at 16:39

Calvin Lin

68,864

This hint makes the question laughable, if it truly did mean '1 as a value' and not 'one value' – MNIShaurya Apr 04 '20 at 16:51

Problem related to polynomial which takes the same value for three inputs

2 Answers2