Polynomial root

Algebra Level pending

Let $p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0$ , where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, then hich of the following is true

p(x)

has one integer roots

p(x)

has all negative roots

p(x)

has no integer roots

p(x)

has all positive roots

1 solution

Tom Engelsman
Nov 15, 2020

Suppose $p(k) = 0$ , where $k \in \mathbb{Z}$ . Then taking everything $mod 2$ , we get the either $p(1) \equiv 0 (mod 2)$ or $p(0) \equiv 0 (mod 2)$ since if $k \equiv 0 (mod 2) \Rightarrow p(k) \equiv p(0) (mod 2)$ . Likewise, if $k \equiv 1 (mod 2) \Rightarrow p(k) \equiv p(1) (mod 2).$ However, $p(1) \equiv p(0) = 1 (mod 2) \Rightarrow$ CONTRADICTION. Thus there are no integral roots of $p(x).$

$\mathbb{Q.} \mathbb{E.} \mathbb{D.}$

Polynomial root

1 solution

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