We know that a quadratic equation has at most two real roots. Now, how many real roots can a cubic equation $x^3 + bx^2 + cx + d = 0$ have? Explain your answer.
I know by the back of my head that a cubic equation has either one real root or three real roots. However, how do I go about proving it? If it's possible, I would appreciate examples to showcase this.
What i have attempted so far:
Since $x^3 + bx^2 + cx + d = 0$ is a cubic polynomial equaation, it is continuous on $[a,b]$, where $a<b$, and differentiable on $(a,b)$. Thus, by Roelle's Theorem, there exists $d ϵ (a,b)$ such that $$ f'(d) = 0 $$ $$ 3x^2 + 2bx + c = 0 $$ Hence, this shows that there exists at least one real root on this cubic equation.
How do I then show it has three real roots as well?