Relations Between the Roots and the Coefficients

Write the general monic degree-$n$ equation as

and let $\alpha_1,\alpha_2,\dots,\alpha_n$ be its roots. Comparing this with the factored form $(x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_n)$ and expanding, the coefficient of each power of $x$ turns out to be (up to a sign) one of the elementary combinations of the roots. Writing $s_k$ for the sum of all products of the roots taken $k$ at a time, the pattern is

In words: the signs of $s_1, s_2, s_3, \dots$ alternate starting from a minus sign, and each $s_k$ equals $\pm p_k$. For a cubic $x^3+p_1x^2+p_2x+p_3=0$ with roots $\alpha,\beta,\gamma$ this reads $\alpha+\beta+\gamma=-p_1$, $\alpha\beta+\beta\gamma+\gamma\alpha=p_2$, $\alpha\beta\gamma=-p_3$ — the same relations you already used for quadratics, just extended one degree further.

These relations are powerful in both directions. Given an equation, you can read off sums and products of the roots without solving it. Given the roots (or some of them), you can reconstruct the equation, because $x^n+p_1x^{n-1}+\cdots+p_n \equiv (x-\alpha_1)\cdots(x-\alpha_n)$ as polynomials, so simply multiplying out the factors gives the coefficients directly.

Worked example. If $2, -1, 3, -2$ are the roots of $x^4 + ax^3 + bx^2 + cx + d = 0$, find $a,b,c,d$. …