Polynomial - Revision history

Lfox at 20:29, 18 November 2024

2024-11-18T20:29:31Z

← Older revision		Revision as of 20:29, 18 November 2024
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	== Examples ==		== Examples ==
	The [[Solid\|volume]] of a cube is a polynomial in its side length.		The [[Solid\|volume]] of a cube is a polynomial in its side length.

			The signed area of an n-dimensional parallelogram spanned by vectors <math>v_1, \cdots, v_n</math> is a polynomial in the components of the vectors. In fact, it is the ''determinant'' of the matrix built out of those components.

	Every [[linear transformation]] is uniquely associated to a polynomial called its characteristic polynomial.		Every [[linear transformation]] is uniquely associated to a polynomial called its characteristic polynomial.

Lfox: /* Examples */

2024-11-18T20:03:04Z

Examples

← Older revision		Revision as of 20:03, 18 November 2024
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	Every [[linear transformation]] is uniquely associated to a polynomial called its characteristic polynomial.		Every [[linear transformation]] is uniquely associated to a polynomial called its characteristic polynomial.

	A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial ~~at all~~ can be realized in this way.		A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial with <math>\mathbb{Z}_{\geq 0}</math> coefficients can be realized in this way.

	Some generating functions in combinatorics or representation theory (those which truncate after a finite number of powers).		Some generating functions in combinatorics or representation theory (those which truncate after a finite number of powers).

Lfox: /* Examples */

2024-06-03T09:33:44Z

Examples

← Older revision		Revision as of 09:33, 3 June 2024
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	A truncated Taylor series approximation to anything. Tangent line to a curve at a point, tangent conic to a curve at a point, tangent cubic to a curve at a point, etc.		A truncated Taylor series approximation to anything. Tangent line to a curve at a point, tangent conic to a curve at a point, tangent cubic to a curve at a point, etc.

			Piecewise polynomial functions (a type of ''spline'') are used in computer graphics to make things look smooth.

	== Rational functions [TODO move to its own page] ==		== Rational functions [TODO move to its own page] ==
	There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book.		There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book.

Lfox at 04:31, 23 April 2024

2024-04-23T04:31:19Z

← Older revision		Revision as of 04:31, 23 April 2024
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	A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial at all can be realized in this way.		A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial at all can be realized in this way.

	~~Generating~~ functions in combinatorics.		Some generating functions in combinatorics or representation theory (those which truncate after a finite number of powers).

			A truncated Taylor series approximation to anything. Tangent line to a curve at a point, tangent conic to a curve at a point, tangent cubic to a curve at a point, etc.

	== Rational functions [TODO move to its own page] ==		== Rational functions [TODO move to its own page] ==
	There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book.		There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book.

Lfox at 02:48, 21 April 2024

2024-04-21T02:48:30Z

← Older revision		Revision as of 02:48, 21 April 2024
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	Generating functions in combinatorics.		Generating functions in combinatorics.

			== Rational functions [TODO move to its own page] ==
			There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book.

Lfox at 03:25, 16 April 2024

2024-04-16T03:25:45Z

← Older revision		Revision as of 03:25, 16 April 2024
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	A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial at all can be realized in this way.		A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial at all can be realized in this way.

			Generating functions in combinatorics.

Lfox: Created page with "A '''polynomial''' is any function which obtains its result solely by some combination of: multiplying its inputs together, adding them together, or rescaling them. == Examples == The volume of a cube is a polynomial in its side length. Every linear transformation is uniquely associated to a polynomial called its characteristic polynomial. A computer program consisting of some number of `for` loops (and no recursion or `goto`..."

2024-04-10T19:34:11Z

Created page with "A '''polynomial''' is any function which obtains its result solely by some combination of: multiplying its inputs together, adding them together, or rescaling them. == Examples == The volume of a cube is a polynomial in its side length. Every linear transformation is uniquely associated to a polynomial called its characteristic polynomial. A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>..."

New page

A '''polynomial''' is any [[function]] which obtains its result solely by some combination of: multiplying its inputs together, adding them together, or rescaling them.

== Examples ==
The [[Solid|volume]] of a cube is a polynomial in its side length.

Every [[linear transformation]] is uniquely associated to a polynomial called its characteristic polynomial.

A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial at all can be realized in this way.