Chapter 4.2, Problem 34AYU

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

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

To determine

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

Answer to Problem 34AYU

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

Here $r=-2$, the third row of synthetic division contains only numbers that are alternating positive. Therefore $r=-2$ is the lower bound.

Explanation of Solution

Given:

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

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

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

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 $a_n$.

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

Here $a_0=\text{ 4}$ and $a_n=3$.

Zeros of $3$, $q=\pm 1,\pm 3$.

Zeros of $4$, $p=\pm 1,\pm 2,\pm 4$.

The potential rational zeros of $f=\pm 1,\pm 2,\pm 3,\pm 4,\pm \frac{2}{3},\frac{4}{3}$.

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.

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

The coefficients are 3, $-2$, 1 and 4.

$r$	Coefficient of $q\left(x\right)$	Remainder
1	3, 1, 2	6
$-1$	3, $-2$, 4	2
$-2$	3, $-8$, 20	$-38$

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

Here $r=-2$, the third row of synthetic division contains only numbers that are alternating positive. Therefore $r=-2$ is the lower bound.