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Mathematics · Ch 9 — Theory of Equations

Forming an Equation from Given Roots

9.1.3

Forming an Equation from Given Roots

The relations of the previous section run just as well in reverse: given a prescribed list of numbers α1,α2,…,αn\alpha_1,\alpha_2,\ldots,\alpha_n that we want to be the roots of an equation, the monic equation having exactly these roots is

(x−α1)(x−α2)⋯(x−αn)=0.(x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_n)=0.

Expanding this product is the same as computing the elementary symmetric functions S1,S2,…,SnS_1,S_2,\ldots,S_n of the given numbers and writing

xn−S1xn−1+S2xn−2−S3xn−3+⋯+(−1)nSn=0,x^n-S_1x^{n-1}+S_2x^{n-2}-S_3x^{n-3}+\cdots+(-1)^nS_n=0,

with the signs alternating and starting negative. …

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