Polynomial

A polynomial is a finite sum of monomials with coefficients in a ring. In one variable $X$ over a commutative ring $R$, writeThe 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 polynomialThe 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.

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.