3.3.1 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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(2 + 4 x^2)^2 (x - 1)^3
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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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Expand[ (1 + 2x + y)^3 ]
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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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Collect[ Expand[ (1 + x + 2y + 3z)^3 ], {x, y} ]
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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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Expand[(x + 1)^2 (y + 1)^2, x]
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| This avoids expanding parts which do not contain objects matching b[_]. | |
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Expand[(a[1] + a[2] + 1)^2 (1 + b[1])^2, b[_]]
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| PowerExpand[expr] | expand out  and  in expr |
Expanding powers. Mathematica does not automatically expand out expressions of the form (a b)^c except when c is an integer. In general it is only correct to do this expansion if a and b are positive reals. Nevertheless, the function PowerExpand does the expansion, effectively assuming that a and b are indeed positive reals. | Mathematica 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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| 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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t = 3 + x f[1] + x^2 f[1] + y f[2]^2 + z f[2]^2
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| This collects terms that match f[_]. | |
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Collect[t, f[_]]
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| This applies Factor to each coefficient obtained. | |
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Collect[t, f[_], Factor]
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