Chapter 4.6, Problem 37E

To determine

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

Answer to Problem 37E

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
3	2	0	5
3	0	2	5
1	2	2	5
1	0	4	5

Explanation of Solution

Given information:

Given function,

  $g\left(x\right)=x^5-3x^3+8x-10$

Calculation:

Consider the function $g\left(x\right)=x^5-3x^3+8x-10$ .

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

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

  $\begin{array}{c}g\left(-x\right)={\left(-x\right)}^5-3{\left(-x\right)}^3+8\left(-x\right)-10\\ =-x^5+3x^3-8x-10\end{array}$

The coefficient in $g\left(-x\right)$ have 2 sign change, so $g\left(x\right)$ has two or zero 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
3	2	0	5
3	0	2	5
1	2	2	5
1	0	4	5