Chapter 4.6, Problem 36E

To determine

To find: the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.

Answer to Problem 36E

The possible number of zeros for $g\left(x\right)$are summarized in below table:

Positive real zeros	Negative real zeros	Imaginary zeros	Total zeros
2	3	0	5
2	1	2	5
0	3	2	5
0	1	4	5

Explanation of Solution

Given information:

Given function,

  $g\left(x\right)=x^5-2x^3-x^2+6$

Calculation:

Consider the function $g\left(x\right)=x^5-2x^3-x^2+6$ .

The coefficient in $g\left(x\right)$ have 2 sign change, so $g\left(x\right)$ has two or zero positive real zero.

Now find $g\left(-x\right)$ by substituting $x=-x$ .

  $\begin{array}{c}g\left(-x\right)={\left(-x\right)}^5-2{\left(-x\right)}^3-{\left(-x\right)}^2+6\\ =-x^5+2x^3-x^2+7\end{array}$

The coefficient in $g\left(-x\right)$ have 3 sign change, so $g\left(x\right)$ has three or one negative real zero.

The possible number of zeros for $g\left(x\right)$are summarized in below table:

Positive real zeros	Negative real zeros	Imaginary zeros	Total zeros
2	3	0	5
2	1	2	5
0	3	2	5
0	1	4	5