Chapter 6.4, Problem 6CYU

To determine

To find: the consecutive value of x where the zeroes of the function is located and graph the function.

Answer to Problem 6CYU

The zeroes of the function lied in $\text{x=-1}$ and $\left(2,3\right)$ .

Explanation of Solution

Given:

  $\text{f}\left(\text{x}\right){\text{=-x}}^{\text{4}}{\text{+x}}^{\text{3}}{\text{+2x}}^{\text{2}}\text{+x+1}$ .

Concept used:

The zeroes of the function resembled the function equates to zero.

  $\text{y=f}\left(\text{x}\right)\text{=0}$ .

The zero normally lies in between negative number and positive number.

Therefore, the consecutive number will negative approaches zero and positive approaches to zero except $\text{y=0}$ itself.

The graph of the function $\text{f}$ is the set of all point in the plane of the form $\left(\text{x,f}\left(\text{x}\right)\right)$ .

The graph can be defined by the graph of $\text{f}$ and the equation for the graph will be $\text{y=f}\left(\text{x}\right)$ .

The graph of the function is special case of the graph of an equation.

Calculation:

The function is given as:

  $\text{f}\left(\text{x}\right){\text{=-x}}^{\text{4}}{\text{+x}}^{\text{3}}{\text{+2x}}^{\text{2}}\text{+x+1}$ .

The table for the graph of the function will be:

By putting the value $\text{x}$ the value of $\text{y}$ can be determined.

$\text{x}$	$\text{-2}$	$\text{-1}$	$\text{0}$	$\text{1}$	$\text{2}$	$\text{3}$
$\text{f}\left(\text{x}\right)$	$\text{-17}$	$\text{0}$	$1$	$4$	$3$	$\text{-32}$

The points in the graph so collected will form a graph for the given equation.

The changes in sign from the result indicates that there are zeroes between $\text{x=2}$ and $\text{x=3}$

The other zeroes at $\text{x=-1}$ .

since, the sign of the function changes from negative to positive and at $\text{x=-1}$ here $\text{f}\left(\text{x}\right)\text{=0}$ .

The graph of the given equation by collecting the points will be:

  

From the graph it easily noticed that graph changes it sign from $\text{x=2}$ to $\text{x=3}$ and $\text{x=-1}$

Hence, the zeroes of the function lied in $\text{x=-1}$ and $\left(2,3\right)$ .