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Please help me with Problem 9. I think that the problem has to be written in terms of binomial theorem positive or negative.

Question: Let $f(x) = x^n + a_{n-1}x^{n-1}+\ldots +a_0$ be a polynomial with integer coefficients and whose degree is atleast $2$. Suppose each $a_i \,(0\leq i \leq n-1)$ is of the form: $$a_i = \pm \frac{17!}{r!(17-r)!} \, \, , 1 \leq r \leq 16$$ show that $f(m) \neq 0 \,\forall m \in \mathbb Z$.

S Mukherjee
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1 Answers1

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Hint:

  • The binomial coefficients $\binom{17}{r}$, for $1 \leq r \leq 16$, are multiples of $17$ but not of $17^2$.

  • If $m \in \mathbb Z$ with $f(m) = 0$, then $m$ is a multiple of $17$.

  • If $m \in \mathbb Z$ with $f(m) = 0$, then $-a_0 = m^n + a_{n-1}m^{n-1}+\cdots +a_1 m$.

lhf
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