Chapter 4.6, Problem 35E

To determine

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

Answer to Problem 35E

  $g\left(x\right)$ either has two positive real zero or no positive real zero.

  $g\left(x\right)$ has one negative real zero.

  $g\left(x\right)$ either has two imaginary zeros or no imaginary zero.

Explanation of Solution

Given information:

Given function,

  $g\left(x\right)=x^3-4x^2+8x+7$

Calculation:

Consider the function $g\left(x\right)=x^3-4x^2+8x+7$ .

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 $x=-x$ .

  $\begin{array}{c}g\left(-x\right)={\left(-x\right)}^3-4{\left(-x\right)}^2+8\left(-x\right)+7\\ =-x^3-4x^2-8x+7\end{array}$

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