Difference between revisions of "Vieta's formulas"
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Theorem 14.1.4 (Vieta’s Formula For Higher Degree Polynomials) | Theorem 14.1.4 (Vieta’s Formula For Higher Degree Polynomials) | ||
| − | In a polynomial <math>a_n x^n + a_{ | + | In a polynomial <math>a_n x^n + a_{n-1} x ^ {n-1} ..... a_1 x^{1} + a_0</math> with roots <math>r_1 r_2 r_3 ... r_n </math> |
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| + | the following holds: | ||
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| + | <cmath>r_1 + r_2 + r_3 + \cdots + r_n = -\frac{a_{n-1}}{a_n}</cmath> | ||
| + | <cmath>r_1r_2 + r_1r_3 + \cdots + r_{n-1}r_n = \frac{a_{n-2}}{a_n}</cmath> | ||
| + | <cmath>r_1r_2r_3 + r_1r_2r_4 + \cdots + r_{n-2}r_{n-1}r_n = -\frac{a_{n-3}}{a_n}</cmath> | ||
| + | <cmath>\cdots</cmath> | ||
| + | <cmath>r_1r_2r_3 \cdots r_n = (-1)^n \frac{a_0}{a_n}</cmath> | ||
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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. | 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. | ||
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Latest revision as of 01:39, 22 October 2025
Theorem 14.1.4 (Vieta’s Formula For Higher Degree Polynomials)
In a polynomial 
with roots 
the following holds:
![\[r_1 + r_2 + r_3 + \cdots + r_n = -\frac{a_{n-1}}{a_n}\]](http://latex.artofproblemsolving.com/5/9/4/5946ea49e095695a55d6dfdc67ad995b8a47ca74.png)
![\[r_1r_2 + r_1r_3 + \cdots + r_{n-1}r_n = \frac{a_{n-2}}{a_n}\]](http://latex.artofproblemsolving.com/5/1/e/51e7d070f90f9ec4cb41807743da083f52a00a8d.png)
![\[r_1r_2r_3 + r_1r_2r_4 + \cdots + r_{n-2}r_{n-1}r_n = -\frac{a_{n-3}}{a_n}\]](http://latex.artofproblemsolving.com/1/8/9/189774670a7bf185e62f70b71297a03b4c94b7d8.png)
![\[\cdots\]](http://latex.artofproblemsolving.com/c/e/2/ce26e72ef1985f46ac9b2fedbe0879501540c94c.png)
![\[r_1r_2r_3 \cdots r_n = (-1)^n \frac{a_0}{a_n}\]](http://latex.artofproblemsolving.com/e/8/8/e88d3a84f40be17d575652d4a78761d850d6c7be.png)
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.
