# 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 t.
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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 x 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 x.
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This avoids expanding parts which do not contain objects matching b[_].
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 PowerExpand[expr] expand out Null and Null in expr PowerExpand[expr,Assumptions->assum] expand out expr assuming assum

Expanding powers and logarithms.

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

The Wolfram System does not automatically expand out this expression.
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PowerExpand does the expansion, effectively assuming that x and y are positive reals.
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Log is not automatically expanded out.
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PowerExpand does the expansion.
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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 f.
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This collects terms that match f[_].
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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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