Chapter 4.2, Problem 33AYU

In Problems 45-50, find the bounds to the zeros of each polynomial function. Use the bounds to obtain a complete graph of f .

45. $f\left(x\right)\text{ = 2}{x}^3\text{ + }{x}^2\text{ - 1}$

To determine

To find: The bounds to zeros of each polynomial function to complete the graph of $f$.

Answer to Problem 33AYU

Here $r=1$, the third row of synthetic division contains only numbers that are positive. Therefore $r=1$ is the upper bound.

Here $r=-1$, the third row of synthetic division contains only numbers that are alternating positive. Therefore $r=-1$ is the lower bound.

Explanation of Solution

Given:

$f\left(x\right)=2x^3+x^2-1$

The degree of the polynomial is 3. Therefore the number of real zeros by real zero theorem can be at most $=3$.

$f\left(x\right)=2x^3+x^2-1$

Rational zeros theorem provides information about the rational zeros of a polynomial function with integer coefficients.

If $\frac{p}{q}$ in its lowest terms is a rational zero of $f$, then $p$ is a factor of $a_0$ and $q$ is the factor of $a_n$.

$f\left(x\right)=2x^3+x^2-1$

Here $a_0=\text{ <mo>−</mo>1}$ and $a_n=2$.

Zeros of $2$, $q=\pm 1,\pm 2$.

Zeros of $-1$, $p=\pm 1$.

The potential rational zeros of $f=\pm 1,\pm 2,\pm \frac{1}{2}$.

To find the upper bound, start with the smallest positive integer potential rational zero $=\pm 1$.

To find the lower bound start the largest negative integer potential rational zero $=-1$.

Use repeated Synthetic division.

$f\left(x\right)=2x^3+x^2-1$

The coefficients are 2, 1 and $-1$.

$r$	Coefficient of $q\left(x\right)$	Remainder
1	2, 3, 3	2
$-1$	2, $-1$, 1	$-2$

Here $r=1$, the third row of synthetic division contains only numbers that are positive. Therefore $r=1$ is the upper bound.

Here $r=-1$, the third row of synthetic division contains only numbers that are alternating positive. Therefore $r=-1$ is the lower bound.