Chapter 4.1, Problem 56AYU

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 $\left|x\right|$.

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

To determine

To find:

a. Each real zero and its multiplicity.

Answer to Problem 56AYU

a. $\text{Real zeros }=-\sqrt{3},-\sqrt{3},2, 2, 2, 2\_$.

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

$\text{The multiplicity of the polynomial }=2\left(\text{Multiplicity }=\text{ 4}\right)\_$.

Explanation of Solution

Given:

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

The real zeros of the polynomial $f\left(x\right)={\left(x+\sqrt{3}\right)}^2{\left(x-2\right)}^4$.

The degree of first polynomial $=2$ .

The degree of second polynomial $=4$ .

Combined degree $=6$ .

The leading term $a_n=1$.

a. $-\sqrt{3}$, $-\sqrt{3}$, 2, 2, 2, 2.

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

The multiplicity of the polynomial $=2$ (Multiplicity $=4$).

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 56AYU

b. $\text{Zero }-\sqrt{3}\text{has even multiplicity}\text{. Therefore the graph touches the }x\text{-axis}\_$.

$\text{Zero }2\text{has even multiplicity}\text{. Therefore the graph touches the }x\text{-axis}\_$.

Explanation of Solution

Given:

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

The real zeros of the polynomial $f\left(x\right)={\left(x+\sqrt{3}\right)}^2{\left(x-2\right)}^4$.

The degree of first polynomial $=2$.

The degree of second polynomial $=4$.

Combined degree $=6$.

The leading term $a_n=1$.

b. Zero $-\sqrt{3}$ has even multiplicity. Therefore the graph touches the $x\text{-axis}$.

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

To determine

To find:

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

Answer to Problem 56AYU

c. $\text{The maximum number of turning points}=\left(n-1\right)=5\_$.

Explanation of Solution

Given:

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

The real zeros of the polynomial $f\left(x\right)={\left(x+\sqrt{3}\right)}^2{\left(x-2\right)}^4$.

The degree of first polynomial $=2$.

The degree of second polynomial $=4$.

Combined degree $=6$.

The leading term $a_n=1$.

c. The degree of $f\left(x\right)=n=6$.

The maximum number of turning points $=\left(n-1\right)=5$.

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 56AYU

d. $\text{For large values of }x\text{in the positive or negative direction value of }f\text{approaches positive infinity }\underset{x\to \infty }{\lim }f\left(x\right)=\infty \_$.

Explanation of Solution

Given:

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

The real zeros of the polynomial $f\left(x\right)={\left(x+\sqrt{3}\right)}^2{\left(x-2\right)}^4$.

The degree of first polynomial $=2$.

The degree of second polynomial $=4$.

Combined degree $=6$.

The leading term $a_n=1$.

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

Here $a_n=1>0$.

For large values of $x$ in the positive direction or negative direction value of $f$ approaches positive infinity $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$.

Unbounded in the positive direction.