Polynomial function

A function expressible as a finite sum of constant multiples of nonnegative integer powers of its input. Its degree is the highest power with a nonzero coefficient, except for the zero polynomial, whose degree is undefined.

Polynomial: An algebraic expression built from variables and coefficients using addition, subtraction, and multiplication. A polynomial function is obtained by evaluating a polynomial expression at each input.
Polynomial addition: The operation that adds polynomials by combining terms with matching powers. Adding polynomial functions produces another polynomial function.
Polynomial interpolation: The construction of a polynomial that matches prescribed values at specified input points. Interpolation uses polynomial functions to pass exactly through a finite set of data points.
Rational function: A function expressible as the ratio of two polynomials, with a nonzero denominator. Unlike a polynomial function, a rational function can be undefined at inputs that make its denominator zero.
Coefficient: A numerical or symbolic factor multiplying a term in an expression. Each coefficient sets the contribution of one power of the input.
Polynomial multiplication: The operation that multiplies polynomials by distributing products across their terms. Products of polynomial functions remain polynomial functions, with degrees adding for nonzero factors.
Taylor series: A power series whose coefficients are determined by a function’s derivatives at a point. Finite Taylor polynomials approximate smooth functions near a chosen input.
Exponential function: A function in which the input appears in the exponent, commonly written as a^x for positive a not equal to one. Its variable exponent produces growth behavior unlike the fixed integer powers of a polynomial.
Degree of a polynomial: The greatest exponent with a nonzero coefficient in a nonzero polynomial. Degree governs the function’s broad end behavior and maximum number of roots.
Polynomial derivative: The derivative of a polynomial, obtained by multiplying each coefficient by its exponent and lowering that exponent by one. Differentiation turns polynomial functions into simpler polynomial functions and locates stationary points.

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