Chapter 4.1, Problem 60AYU

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 $\left|x\right|$.

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

To determine

To find:

a. Each real zero and its multiplicity.

Answer to Problem 60AYU

a. $\text{The real zeros }=0,\sqrt{3},\sqrt{3}$.

$\text{The multiplicity of the polynomial }=0\left(\text{Multiplicity }=1\right)$.

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

Explanation of Solution

Given:

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

To find the real zeros

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

$4x=0,(x^2-3)=0$

$x=0,x^2=3$

$x=0,\sqrt{3 },\sqrt{3}$

a. The real zeros $\text{= 0}$, $\sqrt{3}$, $\sqrt{3}$.

The multiplicity of the polynomial = 0 (Multiplicity = 1).

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

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 60AYU

b. $\text{Zero }0\text{has odd multiplicity}\text{. Therefore the graph crosses the }x\text{-axis}$.

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

Explanation of Solution

Given:

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

To find the real zeros

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

$4x=0,(x^2-3)=0$

$x=0,x^2=3$

$x=0,\sqrt{3 },\sqrt{3}$

b. Zero 0 has odd multiplicity. Therefore the graph crosses the $x\text{-axis}$.

Zero $\sqrt{3}$ 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 60AYU

c. $\text{The maximum number of turning points}=\left(n-1\right)=2$.

Explanation of Solution

Given:

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

To find the real zeros

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

$4x=0,(x^2-3)=0$

$x=0,x^2=3$

$x=0,\sqrt{3 },\sqrt{3}$

c. The degree of $f\left(x\right)=n=3$.

The maximum number of turning points $=\left(n-1\right)=2$.

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 60AYU

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

Unbounded in the positive direction.

Explanation of Solution

Given:

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

To find the real zeros

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

$4x=0,(x^2-3)=0$

$x=0,x^2=3$

$x=0,\sqrt{3 },\sqrt{3}$

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

Here $a_n=4>0$.

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.