Chapter 5.4, Problem 48HP

To determine

To explain: with reasoning that the cubic function has zeroes between $\text{x=2}$ and $\text{x=3}$ .

Answer to Problem 48HP

There has no zeroes in between $\text{x=2}$ and $\text{x=3}$ .

Explanation of Solution

Given:

  

Concept used:

The zeroes of the function will occur in between negative number and positive number.

The number of degrees equal to the number of roots.

Cubic polynomial have $\text{3}$ degrees and also it has $\text{3}$ roots.

Calculation:

  $\text{f}\left(\text{x}\right){\text{=ax}}^{\text{3}}{\text{+bx}}^{\text{2}}\text{+cx+d}$ .

the given table is:

$\text{x}$	$\text{-2}$	$\text{-1}$	$\text{0}$	$\text{1}$	$\text{2}$	$\text{3}$
$\text{y}$	$4$	$\text{-2}$	$\text{-1}$	$\text{1}$	$\text{-2}$	$\text{-2}$

From the table it should be notice that there are the three changes of sign occurs which indicates that there are zeroes in between $\text{x=-2 and x=-1}$ ,between $\text{x=0 and x=1}$ and between $\text{x=1 and x=2}$ .

The function has degree $\text{3}$ ,so it has at-most $\text{3}$ real zeroes and its already found $\text{3}$ zeroes between $\text{x=-2 to x=-1}$ , $\text{x=0 to x=1}$ and $\text{x=1 to x=2}$ .

Hence, there is no zeroes in between $\text{x=2}$ and $\text{x=3}$ .