Chapter 4.2, Problem 37AYU

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

49. $f\left(x\right)\text{ = }{x}^4\text{ + 3}{x}^3\text{ - 5}{x}^2\text{ + 9}$

To determine

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

Answer to Problem 37AYU

Here $r=\text{ 1}$, the third row of synthetic division contains only numbers that are positive. Therefore $r=\text{ 1}$ is the upper bound. There can be no zeros greater than 1.

Here $r=-2$, the third row of synthetic division contains only numbers that are alternating positive. Therefore $r=-2$ is the lower bound. There can be no zeros lesser than $-2$.

Explanation of Solution

Given:

$f\left(x\right)=x^4+3x^3-5x^2+9$

The degree of the polynomial is 4. Therefore the number of real zeros by real zero theorem can be at most $=4$.

$f\left(x\right)=x^4+3x^3-5x^2+9$

Rational zeros theorem provides information about the potential 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 $a_n$.

$f\left(x\right)=x^4+3x^3-5x^2+9$

Here $a_0=\text{ 9}$ and $a_n=1$.

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

Zeros of 9, $p=\pm 1,\pm 3,\pm 9$.

The potential rational zeros of $f=\pm 1,\pm 3,\pm 9$.

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

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

Use repeated Synthetic division,

$f\left(x\right)=x^4+3x^3-5x^2+9$

The coefficients are 1, 3, $-5$, 0 and 9.

$r$	Coefficient of $q\left(x\right)$	Remainder
1	1, 4, $-1$, $-1$	8
$-1$	1, 2, $-7$, 7	2
$-2$	1, 1, $-7$, 14	$-19$

Here $r=1$, the third row of synthetic division contains only numbers that are positive. Therefore $r=1$ is the upper bound. There can be no zeros greater than 1.

Here $r=-2$, the third row of synthetic division contains only numbers that are alternating positive. Therefore $r=-2$ is the lower bound. There can be no zeros lesser than $-2$.