Chapter 3.4, Problem 107E

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.

13. P(x) = 2x⁴ − 9x³ + 9x² + x − 3

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 107E

All possible zeros of polynomial $P\left(x\right)=2x^4-9x^3+9x^2+x-3$ are $\pm 1$, $\pm \frac{1}{2}$, $\pm 3$, $\pm \frac{3}{2}$.

Explanation of Solution

Given:

The polynomial is, $P\left(x\right)=2x^4-9x^3+9x^2+x-3$.

Theorem 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 $3$.

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

Factors of the constant coefficient $3$ are $\pm 1$, $\pm 3$.

All possible rational zeros of the polynomial $P\left(x\right)=2x^4-9x^3+9x^2+x-3$ are $\pm \frac{1}{1}$, $\pm \frac{1}{2}$, $\pm \frac{3}{1}$. $\pm \frac{3}{2}$.

These zeros can also be written as $\pm 1$, $\pm \frac{1}{2}$, $\pm 3$, $\pm \frac{3}{2}$.

Graph of the polynomial $P\left(x\right)=2x^4-9x^3+9x^2+x-3$ is shown below.

Figure 1

The graph clearly shows the intercepts of polynomial on $x$ axis are at $x=-0.5$, $x=1$ and $x=3$.

Thus, the correct rational roots of polynomial $P\left(x\right)=2x^4-9x^3+9x^2+x-3$ are $-\frac{1}{2}$, $+1$ and $+3$.

Conclusion:

All possible zeros of a polynomial $P\left(x\right)=2x^4-9x^3+9x^2+x-3$ using Rational Zeros Theorem are $\pm 1$, $\pm \frac{1}{2}$, $\pm 3$, $\pm \frac{3}{2}$..and from the graph of polynomial, correct real roots of the polynomial are $-\frac{1}{2}$, $+1$ and $+3$.