# Structural Operations on Polynomials

Expand[poly] | expand out products and powers |

Factor[poly] | factor completely |

FactorTerms[poly] | pull out any overall numerical factor |

FactorTerms[poly,{x,y,...}] | pull out any overall factor that does not depend on x, y, ... |

Collect[poly,x] | arrange a polynomial as a sum of powers of x |

Collect[poly,{x,y,...}] | arrange a polynomial as a sum of powers of x, y, ... |

Structural operations on polynomials.

Here is a polynomial in one variable.

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Expand expands out products and powers, writing the polynomial as a simple sum of terms.

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Factor performs complete factoring of the polynomial.

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FactorTerms pulls out the overall numerical factor from

.

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There are several ways to write any polynomial. The functions Expand, FactorTerms, and Factor give three common ways. Expand writes a polynomial as a simple sum of terms, with all products expanded out. FactorTerms pulls out common factors from each term. Factor does complete factoring, writing the polynomial as a product of terms, each of as low degree as possible.

When you have a polynomial in more than one variable, you can put the polynomial in different forms by essentially choosing different variables to be "dominant". Collect[poly, x] takes a polynomial in several variables and rewrites it as a sum of terms containing different powers of the "dominant variable" x.

Here is a polynomial in two variables.

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Collect reorganizes the polynomial so that

is the "dominant variable".

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If you specify a list of variables,

Collect will effectively write the expression as a polynomial in these variables.

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Expand[poly,patt] | expand out poly, avoiding those parts which do not contain terms matching patt |

Controlling polynomial expansion.

This avoids expanding parts which do not contain

.

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This avoids expanding parts which do not contain objects matching

.

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Expanding powers and logarithms.

*Mathematica* does not automatically expand out expressions of the form except when is an integer. In general it is only correct to do this expansion if and are positive reals. Nevertheless, the function PowerExpand does the expansion, effectively assuming that and are indeed positive reals.

*Mathematica* does not automatically expand out this expression.

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PowerExpand does the expansion, effectively assuming that

and

are positive reals.

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Log is not automatically expanded out.

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PowerExpand returns a result correct for the given assumptions.

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Collect[poly,patt] | collect separately terms involving each object that matches patt |

Collect[poly,patt,h] | apply h to each final coefficient obtained |

Ways of collecting terms.

Here is an expression involving various functions

.

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This collects terms that match

.

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This applies

Factor to each coefficient obtained.

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HornerForm[expr,x] | puts expr into Horner form with respect to x |

Horner form.

*Horner form* is a way of arranging a polynomial that allows numerical values to be computed more efficiently by minimizing the number of multiplications.

This gives the Horner form of a polynomial.

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