Chapter 3.4, Problem 100E

To determine

All possible rational zeroes of a given polynomial using Rational Zeros Theorem.

Answer to Problem 100E

All possible zeros of polynomial $Q\left(x\right)=x^4-3x^3-6x+8$ are $\pm 1$, $\pm 2$, $\pm 4$, $\pm 8$.

Explanation of Solution

Given:

The polynomial is, $Q\left(x\right)=x^4-3x^3-6x+8$.

Definition used:

Rational Zeros Theorem states that if the polynomial $P\left(x\right)$ has integer coefficients, then every rational zero of $P\left(x\right)$ is of the form $\frac{p}{q}$ where a factor of the constant coefficient is $p$ and factor of leading coefficient $q$.

Calculation:

The leading coefficient of polynomial is $1$ and the constant coefficient is 8.

The factors of leading coefficient $1$ are $\pm 1$.

Factors of the constant coefficient $8$ are $\pm 1$, $\pm 2$, $\pm 4$, $\pm 8$.

All possible rational zeroes of the polynomial $Q\left(x\right)=x^4-3x^3-6x+8$ are $\pm \frac{1}{1}$, $\pm \frac{2}{1}$, $\pm \frac{4}{1}$, $\pm \frac{8}{1}$, or $\pm 1$, $\pm 2$, $\pm 4$, $\pm 8$.