Polynomial: Difference between revisions
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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. | ||
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 | 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). | |||
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] == | |||
There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book. | |||
Latest revision as of 20:29, 18 November 2024
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.
The signed area of an n-dimensional parallelogram spanned by vectors 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.
A computer program consisting of some number of for loops (and no recursion or gotos or other loops), where each for loop runs exactly times, will take a number of steps to run which is given by some polynomial in . In fact, any polynomial with coefficients can be realized in this way.
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.
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]
There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book.