What is the possible number of positive and negative real roots or zeroes in f (x) = x^3+x^2-4

This is a third degree polynomial, so we know that the maximum amount of roots will be 3.

We also know that imaginary roots come in pairs, so there is either 0 or 2 imaginary roots.

We can tell the amount of positive / negative roots by applying Descartes' rule of signs.

Let's examine the polynomial: x^3 + x^2 - 4

Note the sign changes.

From left to right, there is one sign change.

We now know there is only one positive real root.

Now, we can replace all values of x with - 1 and then simplify. The number of sign changes will tell us the amount of negative roots.

(-1) ^3 + (-1) ^2 - 4

-1 + 1 - 4

From left to right, there are two sign changes, which tells us there will be 2 or 0 negative roots.

If there are 2, they are not imaginary.

If there are 0, they are imaginary.

So, our guaranteed root is:1 positive, real root. And our possible roots are:2 negative, real roots. 0 negative, real roots, but 2 imaginary roots.