Chapter 3.4, Problem 108E

Possible Rational Zeros A polynomial function P and its graph are given. (a) List all possible rational zeros of P given by the Rational Zeros Theorem. (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros.

14. P(x) = 4x⁴ − x³ − 4x + 1

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 108E

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

Explanation of Solution

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.

The given polynomial is, $P\left(x\right)=4x^4-x^3-4x+1$.

The leading coefficient of polynomial is $4$ and the constant coefficient is $1$.

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

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

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

These zeros can also be written as $\pm 1$, $\pm \frac{1}{2}$, $\pm \frac{1}{4}$.

Graph of the polynomial $P\left(x\right)=4x^4-x^3-4x+1$ is shown below.

Figure.1

The graph clearly shows the intercepts of polynomial on $x$ axis are at $x=0.25$ and $x=1$.

So the correct rational roots of polynomial $P\left(x\right)=4x^4-x^3-4x+1$ are $+\frac{1}{4}$ and $+1$.

All possible zeros of a polynomial $P\left(x\right)=4x^4-x^3-4x+1$ using Rational Zeros Theorem are $\pm 1$, $\pm \frac{1}{2}$, $\pm \frac{1}{4}$. and from the graph of polynomial, correct real roots of the polynomial are $+\frac{1}{4}$ and $+1$.