Mathematics · Ch 9 — Theory of Equations

Relations between Roots and Coefficients

9.1.2

Relations between Roots and Coefficients

Suppose $\alpha,\beta,\gamma$ are the three roots of the cubic $x^3+px^2+qx+r=0$ . Since a monic polynomial with these roots must equal $(x-\alpha)(x-\beta)(x-\gamma)$ , we can expand this product and compare it, coefficient by coefficient, with $x^3+px^2+qx+r$ :

(x-\alpha)(x-\beta)(x-\gamma)=x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta\gamma+\gamma\alpha)x-\alpha\beta\gamma.

Matching coefficients gives the three relations

S_1=\alpha+\beta+\gamma=-p,\qquad S_2=\alpha\beta+\beta\gamma+\gamma\alpha=q,\qquad S_3=\alpha\beta\gamma=-r.

The same expansion carried out one degree higher, for the biquadratic $x^4+px^3+qx^2+rx+s=0$ with roots $\alpha,\beta,\gamma,\delta$ , gives

S_1=\sum\alpha=-p,\quad S_2=\sum\alpha\beta=q,\quad S_3=\sum\alpha\beta\gamma=-r,\quad S_4=\alpha\beta\gamma\delta=s,

where each $S_k$ is the sum of all products of the roots taken $k$ at a time. In general, for any monic equation of degree $n$ ,

x^n+p_1x^{n-1}+p_2x^{n-2}+\cdots+p_n=0,

with roots $\alpha_1,\ldots,\alpha_n$ , the pattern continues: $S_k=(-1)^kp_k$ for every $k$ from $1$ to $n$ . …

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