Chapter 4.6, Problem 38E

To determine

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

Answer to Problem 38E

The possible numbers 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,

  $g\left(x\right)=x^5+7x^4-4x^3-3x^2+9x-15$

Calculation:

Consider the function $g\left(x\right)=x^5+7x^4-4x^3-3x^2+9x-15$ .

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 $x=-x$ .

  $\begin{array}{c}g\left(-x\right)={\left(-x\right)}^5+7{\left(-x\right)}^4-4{\left(-x\right)}^3-3{\left(-x\right)}^2+9\left(-x\right)-15\\ =-x^5+7x^4+4x^3-3x^2-9x-15\end{array}$

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 numbers 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