Chapter 4, Problem 18RE

To determine

To analyze: the given polynomial function.

Answer to Problem 18RE

  

Explanation of Solution

Given:

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

Explanations:

Let us consider the following polynomial function.

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

analyze the graph of the given function.

For this follow the following steps:

Step $1$ :

Determination of the end behavior of the graph of the function

The end behavior of the graph of a function is similar to the leading term of the function. So, first of all determine the degree of the function.

So,

  $\begin{array}{l}f\left(x\right)=\left(x-4\right){\left(x+2\right)}^2\left(x-2\right)\\ =\left(x-4\right)\left(x-2\right){\left(x+2\right)}^2\\ =\left(x^2-4x-2x+8\right)\left(x^2+4x+4\right)\\ =\left(x^2-6x+8\right)\left(x^2+4x+4\right)\end{array}$

Further solving,

  $\begin{array}{l}\left(x^2-6x+8\right)\left(x^2+4x+4\right)=x^4-6x^3+8x^2+4x^3-24x^2+32x+4x^2-24x+32\\ =x^4-2x^3-12x^2+8x+32\end{array}$

Here, the degree of the polynomial is $4$ . So, the graph of $f$ behaves like $y=x^4$ for large values of $\left|x\right|$ .

Step $2$ :

Graph the function using a graphing utility.

Using graphing utility software, plot the given function.

  

Step $3$ :

Next, approximate $x$ and the $y$ intercepts of the graph

From the graph, the graph passes through $y$ axis at a single point, so the $y-\mathrm{int}ercept$ is at $y=32$ .

Also, the graph intersects $x-axis$ at two points $x=2$ and $x=4$ . Also, it touches the axis at $x=-2$ .

Step $4$ :

Create a table to find points on the graph around each $x-\mathrm{int}ercept$ .

  

Here, for $x=-2$ , the multiplicity is $2$ , as the $f$ touches the axis at this point.

Step $5$ :

Approximately the turning points of the graph.

From the graph of $f$ , $f$ has three turning points.

Using MAXIMUM, one turning point is at $\left(0.39,32.93\right)$ ,using MINIMUM, the other turning points at $\left(3.34,-25.07\right)$ and $\left(-1,0\right)$ .

Step $6$ :

Complete the graph using all the above information.

  

Step $7$ ;

Determine the range and domain of the function.

Domain and range of the function are the set of all real numbers. So,

  $\begin{array}{l}Domain=\left(-\infty ,\infty \right)\\ Range=\left(-\infty ,\infty \right)\end{array}$

Step $8$ :

Determine where the function is increasing and where it is decreasing.

From the graph, the function is increasing in the interval $\left(-1,0.39\right)$ and $\left(3.34,\infty \right)$ . The function is decreasing in the interval $\left(-\infty ,-1\right)$ and $\left(0.39,3.34\right)$

Conclusion:

Therefore,the given polynomial function is analyzed.