Roots of polynomial equations

For any polynomial equation, the coefficients are intrinsically linked to the sums and products of its roots. Vieta's formulas provide the exact expressions for these links. The key is to look at the coefficients in order, divide by the leading coefficient, and apply an alternating sign pattern.

The real power of Vieta's formulas is in finding the values of other symmetric expressions—expressions which remain unchanged if you swap the roots around. A classic example is finding the sum of the squares of the roots, $\Sigma \alpha^2$. We can't get this directly from Vieta's formulas, but we can derive it using them.

Consider the expansion of $(\Sigma \alpha)^2$. For a cubic with roots $\alpha, \beta, \gamma$: $(\alpha+\beta+\gamma)^2 = \alpha^2+\beta^2+\gamma^2 + 2(\alpha\beta+\beta\gamma+\gamma\alpha)$ Rearranging gives the crucial identity: $\Sigma \alpha^2 = (\Sigma \alpha)^2 - 2(\Sigma \alpha\beta)$

Another key skill is to find a new polynomial equation whose roots are related to the roots of an original equation. For example, if the original equation has roots $\alpha, \beta, \gamma$, we might want to find the equation with roots $2\alpha, 2\beta, 2\gamma$ or $\alpha+3, \beta+3, \gamma+3$. The method involves a clever substitution.