= Vieta formulas for a cubic
{c}
Take the stipulated <polynomial roots> $-2,1,5$. Their first three <elementary symmetric polynomials> are
$$
e_1=-2+1+5=4,\qquad
e_2=(-2)(1)+(-2)(5)+(1)(5)=-7,\qquad
e_3=(-2)(1)(5)=-10.
$$
Their <monic polynomial> is therefore
$$
\boxed{(x+2)(x-1)(x-5)=x^3-4x^2-7x+10.}
$$
The constant <coefficient> is $10$, while the <product> of the three <roots of a polynomial> is $-10$: for a <cubic polynomial>, these have opposite signs. Expanding the factors directly verifies the same <polynomial identity>.
\Image[/vieta-cubic-roots-and-coefficient-ledger.png]
{title=The roots $-2,1,5$ determine the signed coefficients of a cubic}
{description=An original exact calculation: the graph shows the stipulated <polynomial>, and the ledger groups choices of one, two or three root positions into <elementary symmetric polynomials>. The alternating factors $(-1)^k$ convert these values into <monic polynomial> <coefficients>.}
For the nonmonic <polynomial> $-3x^3+12x^2+21x-30$, the same ratios give <sum> $-12/(-3)=4$, pairwise-product sum $21/(-3)=-7$, and <product> $-(-30)/(-3)=-10$. It is $-3$ times the <monic polynomial> above, so its <roots of a polynomial> are the same. Checking only the <sum> and <product> would leave the middle <elementary symmetric polynomial> unchecked; all three are needed to recover a <monic polynomial> of <polynomial degree> three.
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