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.
<h3>So, our guaranteed root is:</h3><h3>1 positive, real root.</h3><h3>And our possible roots are:</h3><h3>2 negative, real roots.</h3><h3>0 negative, real roots, but 2 imaginary roots.</h3>
