Chapter 4.1, Problem 52AYU

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 $\left|x\right|$.

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

To determine

To find:

a. Each real zero and its multiplicity.

Answer to Problem 52AYU

a. $3,-2,2,-2,2,-2,2$

$\text{The multiplicity of the polynomial }=3\left(\text{Multiplicity }=1\right)$.

$\text{The multiplicity of the polynomial }=-2\left(\text{Multiplicity }=\text{ 3}\right)$.

$\text{The multiplicity of the polynomial }=2\left(\text{Multiplicity }=\text{ 3}\right)$.

Explanation of Solution

Given:

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

$f\left(x\right)=2\left(x-3\right){\left(x^2+4\right)}^3=2\left(x-3\right)\left(x^6+12x^4+48x^2+64\right)$

$=2\left(x^7-3x^6+12x^5-36x^4+48x^3-144x^2+64x-192\right)$

a. The real zeros of the polynomial.

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

$3,-2,2,-2,2,-2,2$

The multiplicity of the polynomial $=3\left(\text{Multiplicity }=1\right)$

The multiplicity of the polynomial $=-2\left(\text{Multiplicity }=3\right)$

The multiplicity of the polynomial $=2\left(\text{Multiplicity }=3\right)$

To determine

To find:

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

Answer to Problem 52AYU

b. $\text{Zero 3 has odd multiplicity. Therefore the graph crosses the }x\text{-axis}$.

$\text{Zero }-2\text{ has odd multiplicity. Therefore the graph crosses the }x\text{-axis}$.

$\text{Zero }2\text{ has odd multiplicity. Therefore the graph crosses the }x\text{-axis}$.

Explanation of Solution

Given:

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

$f\left(x\right)=2\left(x-3\right){\left(x^2+4\right)}^3=2\left(x-3\right)\left(x^6+12x^4+48x^2+64\right)$

$=2\left(x^7-3x^6+12x^5-36x^4+48x^3-144x^2+64x-192\right)$

b. Zero $3$ has odd multiplicity. Therefore the graph crosses the $x\text{-axis}$.

Zero $-2$ has odd multiplicity. Therefore the graph crosses the $x\text{-axis}$.

Zero $2$ has odd multiplicity. Therefore the graph crosses the $x\text{-axis}$.

To determine

To find:

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

Answer to Problem 52AYU

c. $\text{The degree of }f\left(x\right)=n=\text{ 7}\_$.

$\text{The maximum number of turning points }=\left(n-1\right)=6\_$.

Explanation of Solution

Given:

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

$f\left(x\right)=2\left(x-3\right){\left(x^2+4\right)}^3=2\left(x-3\right)\left(x^6+12x^4+48x^2+64\right)$

$=2\left(x^7-3x^6+12x^5-36x^4+48x^3-144x^2+64x-192\right)$

c. The degree of $f\left(x\right)=n=7$.

The maximum number of turning points $=\left(n-1\right)=6$.

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 52AYU

d. $\text{For large values of }x\text{ in the positive direction value of }f\text{ approaches positive infinity}$.

$\underset{x\to \infty }{\lim }f\left(x\right)=\infty$

Explanation of Solution

Given:

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

$f\left(x\right)=2\left(x-3\right){\left(x^2+4\right)}^3=2\left(x-3\right)\left(x^6+12x^4+48x^2+64\right)$

$=2\left(x^7-3x^6+12x^5-36x^4+48x^3-144x^2+64x-192\right)$

d. $2\left(x^7-3x^6+12x^5-36x^4+48x^3-144x^2+64x-192\right)$

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

Here $a_n=2>0$.

For large values of $x$ in the positive direction value of $f$ approaches positive infinity.

$\underset{x\to \infty }{\lim }f\left(x\right)=\infty$