📋 Prerequisites

• Quadratics & Parabolas
• Polynomials & Factoring

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. The magic is: you can answer questions about roots without ever finding the roots themselves. This is a competition staple — half the time they ask for $r_1^2 + r_2^2$ or $\frac{1}{r_1} + \frac{1}{r_2}$, you don't need to solve at all.

📚 Key Concepts

Vieta for Quadratics

If $ ax^2 + bx + c = 0 $ has roots $r_1$ and $r_2$:

• Sum of roots: $ r_1 + r_2 = -\frac{b}{a} $
• Product of roots: $ r_1 r_2 = \frac{c}{a} $

Intuition: if $ a(x - r_1)(x - r_2) = ax^2 - a(r_1+r_2)x + a(r_1r_2) $, matching coefficients gives you the formulas.

Symmetric Expressions

These are the most common Vieta problems. Express everything in terms of sum ($S = r_1 + r_2$) and product ($P = r_1r_2$):

• $ r_1^2 + r_2^2 = S^2 - 2P $
• $ r_1^3 + r_2^3 = S^3 - 3SP $
• $ \frac{1}{r_1} + \frac{1}{r_2} = \frac{S}{P} $
• $ (r_1 - r_2)^2 = S^2 - 4P $ (this is related to the discriminant!)
• $ |r_1 - r_2| = \sqrt{S^2 - 4P} $

The general strategy: if you can write the expression using only $+$ and $\times$ on the roots, Vieta can compute it.

Vieta for Cubics

If $ ax^3 + bx^2 + cx + d = 0 $ has roots $r_1, r_2, r_3$:

• $ r_1 + r_2 + r_3 = -\frac{b}{a} $
• $ r_1r_2 + r_1r_3 + r_2r_3 = \frac{c}{a} $
• $ r_1r_2r_3 = -\frac{d}{a} $

Pattern: signs alternate starting with negative for the sum. Each formula has one more level of "productness."

General Vieta (Degree n)

For $ a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0 $ with roots $r_1, ..., r_n$:

• Sum of roots: $ \sum r_i = -\frac{a_{n-1}}{a_n} $
• Sum of products (2 at a time): $ \sum_{i
• Product of all roots: $ r_1r_2 \cdots r_n = (-1)^n \frac{a_0}{a_n} $

Signs flip each time you go one level deeper in products.

Constructing Polynomials from Roots

If you know the roots, you can build the polynomial:

• Monomic (leading coefficient = 1): $ (x - r_1)(x - r_2) \cdots (x - r_n) $
• Use Vieta to find coefficients directly without expanding
• If you need integer coefficients and roots are fractions, multiply through by the denominators

Integer Root Problems

When a quadratic has integer coefficients and integer roots:

• The roots multiply to $c$ (if $a = 1$)
• The roots add to $-b$
• Strategy: list factor pairs of $c$, check which pair adds to $-b$
• This is essentially factoring, but Vieta gives you the theoretical framework

✏️ Example Problems