Chapter 3.4, Problem 105E

To determine

All possible rational zeroes of a given polynomial using Rational Zeros Theorem and using graph of polynomial to find correct roots.

Answer to Problem 105E

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

Explanation of Solution

Given:

The polynomial is $P\left(x\right)=5x^3-x^2-5x+1$

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 the given polynomial is $5$ and the constant coefficient is $1$.

The factors of leading coefficient $5$ are $\pm 1$, $\pm 2$, $\pm 5$.

Factors of the constant coefficient $1$ are $\pm 1$.

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

These zeros can also be written as $\pm 1$, $\pm \frac{1}{2}$, $\pm \frac{1}{5}$.

Graph of the polynomial $P\left(x\right)=5x^3-x^2-5x+1$ is shown below.

Figure 1

The graph clearly shows the $x$ intercepts of polynomial at $x=-1$, $x=0.2$ and $x=1$.

So the correct rational roots of polynomial $P\left(x\right)=5x^3-x^2-5x+1$ are $-1$, $\frac{1}{5}$, $1$.

All possible zeros of a polynomial $P\left(x\right)=5x^3-x^2-5x+1$ using Rational Zeros Theorem are $\pm 1$, $\pm \frac{1}{2}$, $\pm \frac{1}{5}$ and from the graph of polynomial, correct real roots of the polynomial are $-1$, $\frac{1}{5}$, $1$.