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Take the stipulated polynomial roots $-2,1,5$. Their first three elementary symmetric polynomials areTheir monic polynomial is thereforeThe 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.

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.

Vieta's formulas connect two descriptions of a polynomial: its coefficients and its roots of a polynomial. They let us read the sum and product of the roots of a polynomial without finding the individual roots of a polynomial, and turn a proposed factorization of polynomials into checkable coefficient identities. The key is that coefficients encode elementary symmetric polynomials, which are unchanged by a permutation of the roots of a polynomial.

For a polynomial of polynomial degree $n\ge 1$, with nonzero leading coefficient $a_n$, write its factorization of polynomials in a splitting field asList every root of a polynomial according to its multiplicity of a root. ThenIn particular, ${\sum }_i{r}_i=-a_{n-1}/a_n$ and ${\prod }_i{r}_i=(-1{)}^n{a}_0/a_n$. With coefficients in the real numbers or complex numbers, the fundamental theorem of algebra ensures that a list of $n$ roots of a polynomial in the complex numbers exists when multiplicity of a root is counted.^[ref]

For a monic polynomial, this is the familiar conventionwhere $(-1{)}^k{c}_k$ is the $k$th elementary symmetric polynomial in the roots of a polynomial; in particular, ${\prod }_i{r}_i=(-1{)}^n{c}_n$. The ratio by $a_n$ in the general statement matters: multiplying every coefficient by the same nonzero scalar preserves both the roots of a polynomial and these ratios.

A short derivation explains every alternating sign. In the product ${\prod }_i(x-r_i)$, a term of polynomial degree $n-k$ arises by choosing $-r_i$ from $k$ distinct factor positions and $x$ from the others. Summing the resulting products gives $(-1{)}^k{e}_k$. Values at different positions may coincide: the selection is of positions, not distinct root values. Matching coefficients gives the formula. This is the distributive property organized by the number of chosen root factors.

A weak composition of an integer $N\ge 0$ into $k\ge 1$ parts is an ordered tuple of nonnegative integers with sum $N$. Zero parts retain their positions: $(0,2,3)$ and $(2,0,3)$ differ. Fixing the length $k$ is essential when applying stars and bars, since inserting extra zero parts creates further tuples. ^[ref]

For a finite list of real numbers $x_i$ and nonnegative weights of positive total $W$, the weighted mean uniquely minimizes the weighted sum of squared deviationsExpanding around ${\bar{x}}_w$ and using ${\sum }_i{w}_i(x_i-{\bar{x}}_w)=0$ givesEvery displacement from the weighted mean increases the objective. This is the simplest one-parameter weighted least squares problem: represent all entries by one common value.

For the weighted values $90$ and $60$ with weights $2$ and $8$,Thus $S(66)=1440$, while $S(75)=2250$. The minimum is defined by this squared-error criterion; a different loss can select a different summary, such as a median under unweighted absolute deviations.

In the average of partial sums $s_0,\dots ,s_N$, the term $a_k$ occurs exactly $N-k+1$ times. Reordering this finite sum gives the displayed identity. Early terms have weights closer to one, and the newest term has weight $1/(N+1)$. The triangular weight expresses the cumulative averaging rule; it is not a claim that one may freely rearrange an infinite series.