Chapter 2, Problem 64RE

To determine

List all the possible rational zeros using rational zero test and to verify the zeros of the polynomial function $f(x)=10x^3+21x^2-x-6$ whether they are in the list of possible rational zeros list using graph.

Answer to Problem 64RE

  $-2,-\frac{3}{5}\text{,}\frac{1}{2}$

Explanation of Solution

Given:

Function: $f(x)=10x^3+21x^2-x-6$

Calculation:

1. Performing rational zero test:

The leading coefficient of given polynomial is 10 and the constant term is -6.

So, possible rational zeros are:

  $\frac{\text{Factors of}-6}{\text{Factors of 10}}=\frac{\pm 1,\pm 2,\pm 3,\pm 6}{\pm 1,\pm 2,\pm 5,\pm 10}=\pm 1,\pm 2,\pm 3,\pm 6,\pm \frac{1}{2},\pm \frac{3}{2},\pm \frac{1}{5},\pm \frac{2}{5},\pm \frac{3}{5},\pm \frac{6}{5},\pm \frac{1}{10},\pm \frac{3}{10}$

2. Verifying the zeros of given function, if they are present in above list or not.

Calculation for graph Consider $f(x)=10x^3+21x^2-x-6$

Values of x	Values of f ( x )
0	-6
1	24
-1	6
2	156
-2	0

By taking different values of x , the graph can be plotted.

Graph:

  

Interpretation:

From above graph, it is clear that, the rational zeros of given polynomial function are

  $x=-2,-\frac{3}{5}\text{and}\frac{1}{2}.$

Also, these zeros are contained in above possible rational zeros list.