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| − | == Introduction ==
| + | #REDIRECT[[Vieta's formulas]] |
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| − | Let <math>P(x)={a_n}x^n+{a_{n-1}}x^{n-1}+\cdots+{a_1}x+a_0</math>,
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| − | where the coefficient of <math>\displaystyle x^{i}</math> is <math>\displaystyle {a}_i</math>. As a consequence of the [[Fundamental Theorem of Algebra]], we can also write <math>P(x)=a_n(x-r_1)(x-r_2)\cdots(x-r_n)</math>, where <math>\displaystyle {r}_i</math> are the roots of <math>\displaystyle P(x)</math>. We thus have that
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| − | <math> a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 = a_n(x-r_1)(x-r_2)\cdots(x-r_n).</math>
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| − | Expanding out the right hand side gives us
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| − | <center><math> a_nx^n - a_n(r_1+r_2+\cdots+r_n)x^{n-1} + a_n(r_1r_2 + r_1r_3 + \cdots + r_{n-1}r_n)x^{n-2} + \cdots + (-1)^na_n r_1r_2\cdots r_n.</math></center>
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| − | We can see that the coefficient of <math>\displaystyle x^k </math> will be the <math> \displaystyle k </math>th [[symmetric sum]]. The <math>k</math>th symmetric sum is just the sum of the roots taken <math>k</math> at a time. For example, the 4th symmetric sum is <math>\displaystyle r_1r_2r_3r_4 + r_1r_2r_3r_5+\cdots+r_{n-3}r_{n-2}r_{n-1}r_n.</math> Notice that every possible [[combination]] of four roots shows up in this sum.
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| − | We now have two different expressions for <math>\displaystyle P(x)</math>. These ''must'' be equal. However, the only way for two polynomials to be equal for all values of <math>\displaystyle x</math> is for each of their corresponding coefficients to be equal. So, starting with the coefficient of <math>\displaystyle x^n </math>, we see that
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| − | <center><math>\displaystyle a_n = a_n</math></center>
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| − | <center><math> a_{n-1} = -a_n(r_1+r_2+\cdots+r_n)</math></center>
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| − | <center><math> a_{n-2} = a_n(r_1r_2+r_1r_3+\cdots+r_{n-1}r_n)</math></center>
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| − | <center><math>\vdots</math></center>
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| − | <center><math>a_0 = (-1)^n a_n r_1r_2\cdots r_n</math></center>
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| − | More commonly, these are written with the roots on one side and the <math>\displaystyle a_i</math> on the other (this can be arrived at by dividing both sides of all the equations by <math>\displaystyle a_n</math>).
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| − | If we denote <math>\displaystyle \sigma_k</math> as the <math>\displaystyle k</math>th symmetric sum, then we can write those formulas more compactly as <math>\displaystyle \sigma_k = (-1)^k\cdot \frac{a_{n-k}}{a_n{}}</math>, for <math>\displaystyle 1\le k\le {n}</math>.
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| − | == See also ==
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| − | * [[Algebra]]
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| − | * [[Polynomials]]
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| − | * [[Newton sums]]
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| − | == Related Links ==
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| − | [http://mathworld.wolfram.com/VietasFormulas.html Mathworld's Article]
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