Chapter 3.4, Problem 101E

Possible Rational Zeros List all possible rational zeros given by the Rational Zeros Theorem (but don’t check to see which actually are zeros).

7. R(x) = 2x⁵ + 3x³ + 4x² − 8

To determine

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

Answer to Problem 101E

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

Explanation of Solution

Given:

The polynomial is $R\left(x\right)=2x^5+3x^3+4x^2-8$.

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 $2$ and the constant coefficient is $8$.

The factors of leading coefficient $2$ are $\pm 1$, $\pm 2$.

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

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

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