Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. This is arguably the most useful tool for polynomial problems in competitions.

The Formulas

For a polynomial $P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0$ with roots $r_1,r_2,\dots ,r_n$:

• Sum of roots: $\sum r_i=-\frac{a_{n-1}}{a_n}$
• Sum of pairs: ${\sum }_{i<j}{r}_i{r}_j=\frac{a_{n-2}}{a_n}$
• Sum of triples: ${\sum }_{i<j<k}{r}_i{r}_j{r}_k=-\frac{a_{n-3}}{a_n}$
• Product of roots: $r_1{r}_2\cdots r_n=(-1{)}^n\frac{a_0}{a_n}$

Quadratic Case ($n=2$)

For $a{x}^2+bx+c=0$ with roots $r_1,r_2$:

1. $r_1+r_2=-b/a$
2. $r_1{r}_2=c/a$

Cubic Case ($n=3$)

For $a{x}^3+b{x}^2+cx+d=0$ with roots $r_1,r_2,r_3$:

1. $r_1+r_2+r_3=-b/a$
2. $r_1{r}_2+r_2{r}_3+r_3{r}_1=c/a$
3. $r_1{r}_2{r}_3=-d/a$

Newton Sums

A powerful extension for computing sums of powers $S_k=r_1^k+\cdots +r_n^k$. For a monic polynomial $P(x)=x^n+c_1{x}^{n-1}+\cdots +c_n$:

• $S_1+c_1=0$
• $S_2+c_1{S}_1+2c_2=0$
• $S_k+c_1{S}_{k-1}+\cdots +c_{k-1}{S}_1+k{c}_k=0$ (for $k\le n$)

Practice Problems