0
$\begingroup$

Problem:

Let $f(x)$ be a quartic polynomial with integer coefficients and four integer roots. Suppose the constant term of $f(x)$ is $6$.

(a) Is it possible for $x=3$ to be a root of $f(x)$?

(b) Is it possible for $x=3$ to be a double root of $f(x)$?

Prove your answers.

What I know:

For a rational number $\frac{p}{q}$ to be a root of a polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, $p$ must divide $a_0$ and $q$ must divide $a_n.$

How to I complete this problem?

  • 2
    Say $f(x)=(x-1)^2(x-2)(x-3)$, to get started. – 2017-02-14
  • 1
    @JenkinsMa: If the roots are $r,s,t,u$, what is the factored form of the quartic? Looking at the factored form, how does the product of the roots relate to $f(0)$? Looking at the standard form, how does $f(0)$ relate to the constant term? – 2017-02-14
  • 1
    (b): No, because $3^2$ would have to divide the constant term. – 2017-02-14

0 Answers 0