Vieta's formulas

Theorem 14.1.4 (Vieta’s Formula For Higher Degree Polynomials) In a polynomial with roots

the following holds:

\begin{align*} r_1 + r_2 + r_3 + \cdots + r_n (the sum of all terms) &= −\frac{a_{n−1}}{a_n} \\ r_1r_2 + r_1r_3 + \cdots + r_{n−1}r_n (the sum of all products of 2 terms) &= \frac{a_{n−2}}{a_n} \\ r_1r_2r_3 + r_1r_2r_4 + \cdots + r_{n−2}r_{n−1}r_n (the sum of all products of 3 terms) &= −\frac{a_{n−3}{a_n} \\ &\vdots \\ r_1r_2r_3 \cdots r_n (the sum of all products of n terms) &= (−1)^n \frac{a_0}{a_n} \end{align*}

Note that the negative and positive signs alternate. When summing the products for odd number of terms, we will have a negative sign otherwise we will have a positive sign.

This can be used in a variety of problems, such as :

Intermediate Level

AIME I 2001/3 AIME I 2014/5 AIME 1996/5 AIME I 2005/8 AIME 1993/5 AIME II 2008/7

Try these out!

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