Chapter 4.1, Problem 57AYU

In Problems 57-68, for each polynomial function:

(a) List each real zero and its multiplicity.

(b) Determine whether the graph crosses or touches the $x\text{-axis}$ at each $x\text{-intercept}$

(c) Determine the maximum number of turning points on the graph.

(d) Determine the end behavior; that is, find the power function that the graph of $f$ resembles for large values of $\left|x\right|$.

$f\left(x\right)=3\left(x^2+8\right){\left(x^2+9\right)}^2$

To determine

To find:

a. Each real zero and its multiplicity.

Answer to Problem 57AYU

a. There are no real zeros of the polynomial.

Explanation of Solution

Given:

$f\left(x\right)=3(x^2+8){(x^2+9)}^2$

a. To find the real zeros.

$f\left(x\right)=0$

$3(x^2+8)=0,{(x^2+9)}^2=0$

$x^2=-8, x^2=-9,x^2=-9$

There are no real zeros of the polynomial.

To determine

To find:

b. To determine whether the graph crosses or touches the $x\text{-axes}$ at each $x\text{-intercept}$.

Answer to Problem 57AYU

b. $\text{The graph neither crosses or touches the}x\text{-axes}$.

Explanation of Solution

Given:

$f\left(x\right)=3(x^2+8){(x^2+9)}^2$

b. The graph neither crosses or touches the $x\text{-axes}$.

To determine

To find:

c. To determine the maximum number of turning points on the graph.

Answer to Problem 57AYU

c. The maximum number of turning points on the graph is 2.

Explanation of Solution

Given:

$f\left(x\right)=3(x^2+8){(x^2+9)}^2$

c. The maximum number of turning points on the graph is $\left(n-1\right)$ is 2.

To determine

To find:

d. To determine the end behavior (power function that the graph of $f$ resembles for large values of $x$).

Answer to Problem 57AYU

d. $\text{The end behavior is }y=3x^6$.

Explanation of Solution

Given:

$f\left(x\right)=3(x^2+8){(x^2+9)}^2$

d. The polynomial $f\left(x\right)=a_n{x}^n$.

In this equation $a_n=3>0$

The end behavior is $y=3x^6$.