Difference between revisions of "Vieta's formulas"
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* [[2007 AMC 12A Problems/Problem 21 | 2007 AMC 12A Problem 21]] | * [[2007 AMC 12A Problems/Problem 21 | 2007 AMC 12A Problem 21]] | ||
* [[2010 AMC 10A Problems/Problem 21 | 2010 AMC 10A Problem 21]] | * [[2010 AMC 10A Problems/Problem 21 | 2010 AMC 10A Problem 21]] | ||
− | * [[2003 AMC 10A Problems/Problem 18 | | + | * [[2003 AMC 10A Problems/Problem 18 | 2003 AMC 10A Problem 18]] |
+ | * [[2021 AMC 12A Problems/Problem 12 | 2021 AMC 12A Problem 12]] | ||
=== Intermediate === | === Intermediate === |
Latest revision as of 20:24, 21 October 2024
In algebra, Vieta's formulas are a set of results that relate the coefficients of a polynomial to its roots. In particular, it states that the elementary symmetric polynomials of its roots can be easily expressed as a ratio between two of the polynomial's coefficients.
It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many math contests/tournaments.
Statement
Let be any polynomial with complex coefficients with roots , and let be the elementary symmetric polynomial of the roots.
Vieta’s formulas then state that This can be compactly summarized as for some such that .
Proof
Let all terms be defined as above. By the factor theorem, . We will then prove Vieta’s formulas by expanding this polynomial and comparing the resulting coefficients with the original polynomial’s coefficients.
When expanding the factorization of , each term is generated by a series of choices of whether to include or the negative root from every factor . Consider all the expanded terms of the polynomial with degree ; they are formed by multiplying a choice of negative roots, making the remaining choices in the product , and finally multiplying by the constant .
Note that adding together every multiplied choice of negative roots yields . Thus, when we expand , the coefficient of is equal to . However, we defined the coefficient of to be . Thus, , or , which completes the proof.
Problems
Here are some problems with solutions that utilize Vieta's quadratic formulas:
Introductory
- 2005 AMC 12B Problem 12
- 2007 AMC 12A Problem 21
- 2010 AMC 10A Problem 21
- 2003 AMC 10A Problem 18
- 2021 AMC 12A Problem 12