Question

What are the possible numbers of positive, negative, and complex zeros of f(x) = -3x4 + 5x3 - x2 + 8x + 4?

Answer 1

Positive Roots: 3 or 1
Negative Roots: 1

Answer 2

3 positive real root and 1 negative real root and no complex root.

Explanation:

Here, the given function,

`f(x) = -3x^4 + 5x^3 - x^2 + 8x + 4`

Since, the coefficient of variables are,

-3, 5, -1, 8, 4,

The sign of variables goes from Negative(-3) to positive (5) , positive(5) to negative(-1) and negative (-1) to positive (8),

So, the total changes in sign = 3,

By the Descartes's rule of sign,

Hence, the number of real positive roots = 3,

Also,

`f(-x) = -3x^4 - 5x^3 - x^2 - 8x + 4`

Since, the coefficient of variables are,

-3, -5, -1, -8, 4

The sign of varibles goes to negative (-8) to positive (4),

So, the total changes in sign = 1,

Hence, the number of real negative roots = 1,

Now, the degree of the function f(x) is 4,

Thus, the highest number of roots of f(x) is 4,

So, it does not have any complex root.