= Polynomial
{wiki}
A polynomial is a finite <sum> of <monomials> with <coefficients> in a <ring>. In one <variable> $X$ over a <commutative ring> $R$, write
$$
p(X)=a_0+a_1X+\cdots+a_nX^n,\qquad a_j\in R.
$$
The symbol $X$ is an <indeterminate>: it records powers formally before we choose an input value. Finite expressions are easy to add, multiply, evaluate, and differentiate, making <polynomials> a common language for exact <algebra> and approximate <numerical analysis>.
Consider the real <polynomial>
$$
p(X)=X^3-3X^2+4=(X+1)(X-2)^2.
$$
The expanded form displays the <coefficients> and the <degree of a polynomial>; the factored form displays its <roots of a polynomial>. At $X=3$, either description gives $p(3)=4$. At $X=-1$ and $X=2$, the factored form makes the zero values immediate. Keeping several descriptions of one object is useful because each makes a different question easy.
\Image[/polynomial-roots-and-multiplicity.png]
{title=A simple <polynomial root> crosses the axis, while a double root touches it; perturbing the double factor can produce real or complex roots}
The graph shows another connection. The factor $X+1$ changes sign across $-1$, while $(X-2)^2$ stays nonnegative across $2$. Thus the graph crosses the axis at the simple <root of a polynomial> and touches it at the double root. <Multiplicity of a root> turns an algebraic repetition into visible local behavior.
For calculations, <Horner's method> evaluates a <polynomial> through nested <multiplication>. For modeling, a <Taylor polynomial> describes local behavior near a point, while <polynomial interpolation> constructs a <polynomial> matching prescribed data. Their usefulness comes with assumptions: a local approximation and an exact identity are different claims, and changing the coefficient <ring> can change which <polynomial roots> exist.
Back to article page
