Chapter 3.4, Problem 104E

To determine

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

Answer to Problem 104E

All possible zeros of polynomial $U\left(x\right)=12x^5+6x^3-2x-8$ are $\pm 1$, $\pm 2$, $\pm 3$, $\pm 4$, $\pm 6$, $\pm 12$, $\pm \frac{1}{2}$, $\pm \frac{3}{2}$, $\pm \frac{1}{3}$, $\pm \frac{2}{3}$, $\pm \frac{4}{3}$, $\pm \frac{1}{6}$, $\pm \frac{2}{3}$.

Explanation of Solution

Given:

The polynomial is $U\left(x\right)=12x^5+6x^3-2x-8$.

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 $p$ is a factor of the constant coefficient and $q$ is a factor of leading coefficient.

Calculation:

The leading coefficient of polynomial is $12$ and the constant coefficient is $8$.

The factors of leading coefficient $12$ are $\pm 1$, $\pm 2$, $\pm 3$, $\pm 4$, $\pm 6$, $\pm 12$.

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

All possible rational zeros of the polynomial, $U\left(x\right)=12x^5+6x^3-2x-8$ are $\pm \frac{1}{1}$, $\pm \frac{1}{2}$, $\pm \frac{1}{3}$, $\pm \frac{1}{4}$, $\pm \frac{1}{6}$, $\pm \frac{1}{12}$, $\pm \frac{2}{1}$, $\pm \frac{2}{2}$, $\pm \frac{2}{3}$, $\pm \frac{2}{4}$, $\pm \frac{2}{6}$, $\pm \frac{2}{12}$ $\pm \frac{4}{1}$, $\pm \frac{4}{2}$, $\pm \frac{4}{3}$, $\pm \frac{4}{4}$, $\pm \frac{4}{6}$, $\pm \frac{4}{12}$ $\pm \frac{8}{1}$, $\pm \frac{8}{2}$, $\pm \frac{8}{3}$, $\pm \frac{8}{4}$, $\pm \frac{8}{6}$, $\pm \frac{8}{12}$.

Eliminate duplicates to get all possible real zeros, $\pm \frac{1}{1}$, $\pm \frac{1}{2}$, $\pm \frac{1}{3}$, $\pm \frac{1}{4}$, $\pm \frac{1}{6}$, $\pm \frac{1}{12}$, $\pm \frac{2}{1}$, $\pm \frac{2}{3}$, $\pm \frac{4}{1}$, $\pm \frac{4}{3}$, $\pm \frac{8}{1}$, $\pm \frac{8}{3}$,these zeros can also be written as $\pm 1$, $\pm \frac{1}{2}$, $\pm \frac{1}{3}$, $\pm \frac{1}{4}$, $\pm \frac{1}{6}$, $\pm \frac{1}{12}$, $\pm 2$, $\pm \frac{2}{3}$, $\pm 4$, $\pm \frac{4}{3}$, $\pm 8$, $\pm \frac{8}{3}.$

All possible zeros of a polynomial $U\left(x\right)=12x^5+6x^3-2x-8$ using Rational Zeros Theorem are $\pm 1$, $\pm \frac{1}{2}$, $\pm \frac{1}{3}$, $\pm \frac{1}{4}$, $\pm \frac{1}{6}$, $\pm \frac{1}{12}$, $\pm 2$, $\pm \frac{2}{3}$, $\pm 4$, $\pm \frac{4}{3}$, $\pm 8$, $\pm \frac{8}{3}.$.