Difference between revisions of "Vieta's formulas"
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− | 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 polynomial | elementary symmetric polynomials]] of its roots can be easily expressed as a | + | 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 polynomial | 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 all mathematics contests. | It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in all mathematics contests. |
Revision as of 20:17, 5 November 2021
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 all mathematics contests.
Statement
Let be any polynomial with complex coefficients with roots , and let be the th elementary symmetric polynomial of the roots. Vietas formulas then state that This can be compactly written as for some such that
Proof
Let all terms be defined as above. By the factor theorem, ; we will then prove these formulas by expanding this polynomial. When expanding this polynomial, every term is generated by choices whether to include or from any factor .
Consider all the expanded terms of with degree ; they are formed by choosing of the negative roots, then by making the remaining choices . Thus, every term is equal to a product of of the negative roots multiplied by . If one factors out , we are left with the th elementary symmetric polynomial of the roots. Thus, when expanding , 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 that test knowledge of Vieta's formulas.