Chapter 2, Problem 9CT

To determine

Find the all rational zeros of the graph and also draw the graph.

Answer to Problem 9CT

The rational zeros of the function is $x=-2,\frac{3}{2}$ .

Explanation of Solution

Given:

It is given in the question that the function is $g(t)=2t^4-3t^3+16t-24$ .

Concept Used:

In this, use the concept of synthetic division and also know that the possible rational zeros are every combination of $\pm \frac{p}{q}$ and also draw the graph.

Calculation:

Here, the function is $g(t)=2t^4-3t^3+16t-24$ .

Factors of constant term $24$ :

  $p=1,2,3,4,6,8,12,24$

Factors of the leading coefficient $2$ :

  $q=1,2$

Possible rational zeros are every combination of $\pm \frac{p}{q}$ .

  $\pm \frac{p}{q}=(1,2,3,4,6,8,12,24,\frac{1}{2},\frac{2}{2},\frac{3}{2},\frac{4}{2},\frac{6}{2},\frac{8}{2},\frac{12}{2},\frac{24}{2})$

Remove duplicate values and simplify

  $\pm \frac{p}{q}=(1,2,3,4,6,8,12,24,\frac{1}{2},\frac{3}{2})$

The graph for the function is given below:

  

The graph seems to indicate $x=-2,\frac{3}{2}$ are the only rational zeros. It can be check with synthetic division.

  $\begin{array}{l}-2\overline{)\begin{array}{l}2-3016-24\\ -414-2824\end{array}}\\ 2-714-120\\ \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\overline{)\begin{array}{l}2-3016-24\\ 30024\end{array}}\\ 200160\end{array}$

The zeros at the bottom right confirm.

Conclusion:

The zeros of the function is $x=-2,\frac{3}{2}$ .