Chapter 5.3, Problem 56PPS

(a)

To determine

To Find: The $x$ and $y$ intercepts, roots, degree, and end behaviour of the given function.

Answer to Problem 56PPS

The x intercept is $x=2,-1,3,-4$ and the y intercept is $24$ . The end function is $g\left(x\right)\to +\infty ,\text{as}x\to -\infty$ and $g\left(x\right)\to -\infty ,\text{as}x\to +\infty$ .

Explanation of Solution

Given:

The given function is $g\left(x\right)=\left(x-2\right)\left(x+1\right)\left(x-3\right)\left(x+4\right)$ .

Calculation:

Consider the given function is,

  $g\left(x\right)=\left(x-2\right)\left(x+1\right)\left(x-3\right)\left(x+4\right)$

Then,

  $\begin{array}{l}\left(x-2\right)\left(x+1\right)\left(x-3\right)\left(x+4\right)\\ x=2,-1,3,-4\end{array}$

Then, the y intercept is,

  $\begin{array}{c}y=\left(0-2\right)\left(0+1\right)\left(0-3\right)\left(0+4\right)\end{array}$

    $\begin{array}{c}=24\end{array}$

Thus, the x intercept is $x=2,-1,3,-4$ and the y intercept is $24$ .

The number of real solutions of the expression $g\left(x\right)$ is four. Then, the required degree is 4 as the degree is even so the end behaviour is,

  $\begin{array}{l}g\left(x\right)\to +\infty ,\text{as}x\to -\infty \\ g\left(x\right)\to -\infty ,\text{as}x\to +\infty \end{array}$

(b)

To determine

To Find: The standard form of the function.

Answer to Problem 56PPS

The standard form of the equation is $x^4-15x^2+10x+24$ .

Explanation of Solution

Consider the given function is,

  $g\left(x\right)=\left(x-2\right)\left(x+1\right)\left(x-3\right)\left(x+4\right)$

Then,

  $\begin{array}{c}g\left(x\right)=\left(x-2\right)\left(x+1\right)\left(x-3\right)\left(x+4\right)\end{array}$

     $\begin{array}{c}=\left(x^2-x-2\right)\left(x^2+x+12\right)\end{array}$

     $\begin{array}{c}=x^4-15x^2+10x+24\end{array}$

(c)

To determine

To Find: The table of the values for the function.

Answer to Problem 56PPS

The table for the value is shown in Table 1

Explanation of Solution

Consider the given function is,

  $g\left(x\right)=\left(x-2\right)\left(x+1\right)\left(x-3\right)\left(x+4\right)$

The table for the value is shown in Table 1

Table 1

$x$	$g\left(x\right)$
-5	224
-4	0
-4	0
-3	-60
-2	-40
-1	0
0	24
1	20
2	0
3	0
4	80
5	324

(d)

To determine

To Find: The sketch for the graph of the function with the plotting points.

Answer to Problem 56PPS

The required graph is shown in Figure 2

Explanation of Solution

From the table 1 the point graph is shown in Figure 1

  

Figure 1