polynomial

Examples of polynomial in the following topics:

• The Method of Partial Fractions

  • Any rational function of a real variable can be written as the sum of a polynomial and a finite number of rational fractions whose denominator is the power of an irreducible polynomial and whose numerator has a degree lower than the degree of this irreducible polynomial.

• Expressing Functions as Power Functions

  • Polynomials are made of power functions.
  • Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
  • Figure shows $\sin x$ and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
  • As more power functions with larger exponents are added, the Taylor polynomial approaches the correct function.

• Taylor Polynomials

  • Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.

• Cylinders and Quadric Surfaces

  • A quadric surface is any $D$-dimensional hypersurface in $(D+1)$-dimensional space defined as the locus of zeros of a quadratic polynomial.
  • A quadric, or quadric surface, is any $D$-dimensional hypersurface in $(D+1)$-dimensional space defined as the locus of zeros of a quadratic polynomial.

• Linear and Quadratic Functions

  • A quadratic function, in mathematics, is a polynomial function of the form: $f(x)=ax^2+bx+c, a \ne 0$.
  • The expression $ax^2+bx+c$ in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of $x$ is 2.

• Applications of Taylor Series

  • The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function.
  • As more terms are added to the Taylor polynomial, it approaches the correct function.
  • This image shows $\sin x$ and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

• Numerical Integration

  • Typically these interpolating functions are polynomials.
  • The simplest method of this type is to let the interpolating function be a constant function (a polynomial of degree zero) which passes through the point $\left(\frac{(a+b)}{2}, f\left(\frac{(a+b)}{2}\right)\right)$.
  • The interpolating function may be an affine function (a polynomial of degree 1) which passes through the points $(a, f(a))$ and $(b, f(b))$.

• Infinite Limits

  • For a rational function $f(x)$ of the form $\frac{p(x)}{q(x)}$, there are three basic rules for evaluating limits at infinity ($p(x)$ and $q(x)$ are polynomials):
  • Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

• Further Transcendental Functions

  • Such a function cannot be expressed as a solution of a polynomial equation whose coefficients are themselves polynomials with rational coefficients.

• The Natural Logarithmic Function: Differentiation and Integration

  • The Taylor polynomials for $\ln(1 + x)$ only provide accurate approximations in the range $-1 < x \leq 1$.
  • Note that, for $x>1$, the Taylor polynomials of higher degree are worse approximations.