Vieta's formulas

For Viète's formula for computing π, see that article.

In mathematics, more specifically in algebra, Viète's formulas, named after François Viète, are formulas which relate the roots of a polynomial to its coefficients.

The formulas

If

$P(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$

is a polynomial of degree $n\geq 1$ with complex coefficients (so the numbers $a_{0},a_{1},\dots ,a_{n-1},a_{n}$ are complex with $a_{n}\neq 0$), by the fundamental theorem of algebra $P(X)$ has $n$ (not necessarily distinct) complex roots $x_{1},x_{2},\dots ,x_{n}.$ Viète's formulas state that

${\begin{cases}x_{1}+x_{2}+\dots +x_{n-1}+x_{n}={\tfrac {-a_{n-1}}{a_{n}}}\\(x_{1}x_{2}+x_{1}x_{3}+\cdots +x_{1}x_{n})+(x_{2}x_{3}+x_{2}x_{4}+\cdots +x_{2}x_{n})+\cdots +x_{n-1}x_{n}={\frac {a_{n-2}}{a_{n}}}\\\vdots \\x_{1}x_{2}\dots x_{n}=(-1)^{n}{\tfrac {a_{0}}{a_{n}}}\end{cases}}$

In other words, the sum of all possible products of $k$ roots of $P(X)$ (with the indices in each product in increasing order so that there are no repetitions) equals $(-1)^{k}a_{n-k}/a_{n},$

$\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}}=(-1)^{k}{\frac {a_{n-k}}{a_{n}}}$

for each $k=1,2,\dots ,n.$

Viète's formulas hold more generally for polynomials with coefficients in any commutative ring, as long as that polynomial of degree $n$ has $n$ roots in that ring.

Example

For the second degree polynomial $P(X)=aX^{2}+bX+c$, Viète's formulas state that the solutions $x_{1}$ and $x_{2}$ of the equation $P(X)=0$ satisfy

$x_{1}+x_{2}=-{\frac {b}{a}},\quad x_{1}x_{2}={\frac {c}{a}}.$

The first of these equations can be used to find the minimum (or maximum) of P. See second order polynomial.

Proof

Viète's formulas can be proved by writing the equality

$a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}=a_{n}(X-x_{1})(X-x_{2})\cdots (X-x_{n})$

(which is true since $x_{1},x_{2},\dots ,x_{n}$ are all the roots of this polynomial), multiplying through the factors on the right-hand side, and identifying the coefficients of each power of $X.$

See also

• Newton's identities
• Elementary symmetric polynomial
• Symmetric polynomial
• Properties of polynomial roots

References

• Vinberg, E. B. (2003). A course in algebra. American Mathematical Society, Providence, R.I. ISBN 0821834134.

• Djukić, Dušan; et al. (2006). The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959-2004. Springer, New York, NY. ISBN 0387242996. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help); Explicit use of et al. in: |first= (help)