Chapter 4, Problem 32CR

To determine

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

Answer to Problem 32CR

The number of positive real zeros, negative real zeros and imaginary zeros are 2 or 0, 0 or 0 and 0 or 4 respectively.

Explanation of Solution

Given information:

The function: $f\left(x\right)=x^4-10x+8$

Calculation:

The given function is $f\left(x\right)=x^4-10x+8$ .

It is seen that the given function has 2 changes of sign.

This means the function $f$ has 2 or 0 positive real zeros.

Also, the given function can be written for negative values of x as:

  $f\left(-x\right)=x^4+10x+8$

It is seen that the above function has 0 changes of sign.

This means the function $f$ has 0 negative real zeros.

The number of positive real zeros, negative real zeros and imaginary zeros is shown in a table below:

Positive real zeros	Negative real zeros	Imaginary zeros	Total zeros
2	0	2	4
0	0	4	4

Hence, the number of positive real zeros, negative real zeros and imaginary zeros are 2 or 0, 0 or 0 and 0 or 4 respectively.