Chapter 4.2, Problem 97AYU

Let $f\left(x\right)$ be a polynomial function whose coefficients are integers. Suppose that $r$ is a real zero of $f$ and that the leading coefficient of $f$ is 1. Use the Rational Zeros Theorem to show that $r$ is either an integer or an irrational number.

To determine

To prove: $r$ is either an integer or an irrational number using Rational zeroes theorem.

Answer to Problem 97AYU

$r$ is either an integer or an irrational number using Rational zeroes theorem.

Explanation of Solution

Given:

Let $f\left(x\right)$ be a polynomial function whose coefficients are integers. Suppose that $r$ is a real zero of f and that the leading coefficient of $f$ is 1.

Proof:

Let $f$ be a polynomial function of degree 1 or higher of the form.

$f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots \dots \mathrm{..}+a_{1}x+a_0, a_n\ne 0,a_0\ne 0$

Where each coefficient is an integer. If $\frac{p}{q}$, in lowest terms, is a rational zero of $f$, then $p$ must be a factor of $a_0$ and $q$ must be a factor of $a_n$.

It is given that $r$ is a real zero of $f$. The rational zeroes of $f$ are integers. So $r$ is either an integer or not an rational number. Hence it is irrational.