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If a nonzero real polynomial has $N$ roots in an open real interval, counted with multiplicity of a root, its derivative has at least $N-1$ roots in that interval, also counted with multiplicity, provided $N\ge 1$. Each root of multiplicity $m$ becomes a root of the derivative of multiplicity $m-1$. The Rolle theorem supplies one further root between each pair of distinct consecutive roots. Thus $k$ distinct roots of multiplicities $m_j$ give at least ${\sum }_j(m_j-1)+(k-1)=N-1$ derivative roots.

A root of a polynomial $f$ is a scalar $a$ satisfying $f(a)=0$ in the chosen field or ring. For $p(X)=(X+1)(X-2{)}^2$ over the real numbers, the distinct roots of a polynomial are $-1$ and $2$. The second has multiplicity of a root two, so there are three roots when counted with multiplicity.

The domain matters. $X^2+1$ has no real polynomial root, but has the two complex number roots $i,-i$. The fundamental theorem of algebra guarantees a polynomial root in the complex numbers for every nonconstant complex polynomial, and complete factorization counts its degree of a polynomial many roots with multiplicity of a root. It does not promise that those roots are real or that they have a simple expression.

The solution set of a mathematical condition $P$ in a specified domain $D$ collects every allowed value that satisfies it:It describes the answer without committing to the steps used to find it. For the equation $x^2=1$ over the real numbers, the solution set is $\{-1,1\}$; restricting to positive real numbers leaves $\{1\}$. For $2x=1$, the solution set is $\{1/2\}$ over the rational numbers and the empty set over the integers. The domain is part of the question.

An equation can have one solution, several solutions, no solutions, or an entire domain of solutions. Over the real numbers, $2x+3=9$ gives the singleton set $\{3\}$, $x^2+1=0$ gives the empty set, and the mathematical identity $(x+1{)}^2=x^2+2x+1$ gives the whole real line. Changing that second equation's domain to the complex numbers gives $\{-i,i\}$. The elementary definition by values that satisfy an equation and the classification by solution sets appear in College Algebra 2e, §2.2: Solving Linear Equations in One Variable.

Requiring several conditions simultaneously takes the intersection of their solution sets. The squared equation $x=(x-2{)}^2$ has solution set $\{1,4\}$. Adding the restriction $x\ge 2$ giveswhich is also the solution set of $\sqrt{x}=x-2$. The extraneous solution example explains why the sign restriction is needed. Requiring at least one of several conditions instead takes their union. This turns logical combinations of constraints into reusable operations on sets.

For several variables, a solution is an ordered assignment, usually represented by a tuple. The real equation $x+y=1$ has solution set $\{(t,1-t):t\in \mathbb{R}\}$, a straight line in the Euclidean plane with coordinates $(x,y)$. Adding $x-y=0$ intersects that straight line with another, leaving the single tuple $(1/2,1/2)$. A system of linear inequalities has no solution exactly when the intersection of its individual solution sets is the empty set. For optimization constraints, this common solution set is the feasible region.

Because a solution set is a set, the order in which its members are listed is irrelevant. The multiple root of $(x-1{)}^2=0$ is still the one solution $1$; its multiplicity of a root is additional polynomial information, rather than a second occurrence inside the solution set. Equivalent equations preserve the solution set on the stated domain, even if their algebraic forms differ.

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 zero of a cyclic code is a polynomial root of its generator polynomial, taken in a splitting field. Every codeword, represented as a polynomial, vanishes there. For a repeated-root cyclic code, the multiplicity of a root must also be retained: ordinary evaluations alone need not determine the code.