Examples of polynomial in the following topics:
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- 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.
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- 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.
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- Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
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- 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.
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- 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.
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- 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.
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- 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 $(b, f(b))$.
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- 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.
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- Such a function cannot be expressed as a solution of a polynomial equation whose coefficients are themselves polynomials with rational coefficients.
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- 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.