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Picking Out Pieces of Algebraic Expressions

Coefficient[expr,form]coefficient of form in expr
Exponent[expr,form]maximum power of form in expr
Part[expr,n] or expr[[n]]nth term of expr

Functions to pick out pieces of polynomials.

Here is an algebraic expression.
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This gives the coefficient of x in e.
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Exponent[expr, y] gives the highest power of y that appears in expr.
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This gives the fourth term in e.
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You may notice that the function Part[expr, n] used to pick out the nth term in a sum is the same as the function described in "Manipulating Elements of Lists" for picking out elements in lists. This is no coincidence. In fact, as discussed in "Manipulating Expressions like Lists", every Mathematica expression can be manipulated structurally much like a list. However, as discussed in "Manipulating Expressions like Lists", you must be careful, because Mathematica often shows algebraic expressions in a form that is different from the way it treats them internally.
Coefficient works even with polynomials that are not explicitly expanded out.
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Numerator[expr]numerator of expr
Denominator[expr]denominator of expr

Functions to pick out pieces of rational expressions.

Here is a rational expression.
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Denominator picks out the denominator.
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Denominator gives 1 for expressions that are not explicit quotients.
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