Vieta's Theorem for Higher-Degree Equations
Big Idea: Vieta's Theorem reveals a beautiful connection between the roots of a polynomial and its coefficients — without ever needing to find the roots themselves!
1. Quick Review: Vieta's for Quadratics
You already know that for a quadratic equation $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$:
$$ r_1 + r_2 = -\frac{b}{a} \qquad \text{and} \qquad r_1 \cdot r_2 = \frac{c}{a} $$
Now we're going to generalize this idea to polynomials of any degree!