Editing Indeterminate (variable) (section)

==Polynomials==
{{main|Polynomial}}
A polynomial in an indeterminate <math>X</math> is an expression of the form <math>a_0 + a_1X + a_2X^2 + \ldots + a_nX^n</math>, where the ''<math>a_i</math>'' are called the [[coefficient]]s of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal.<ref>{{harvnb|Herstein|1975|loc=Section 3.9}}.</ref> In contrast, two polynomial functions in a variable ''<math>x</math>'' may be equal or not at a particular value of ''<math>x</math>''.

For example, the functions
:<math>f(x) = 2 + 3x, \quad g(x) = 5 + 2x</math>

are equal when ''<math>x = 3</math>'' and not equal otherwise. But the two polynomials

:<math>2 + 3X, \quad 5 + 2X</math>
are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact,
:<math>2 + 3X = a + bX</math>

does not hold ''unless'' ''<math>a = 2</math>'' and ''<math>b = 3</math>''. This is because ''<math>X</math>'' is not, and does not designate, a number.

The distinction is subtle, since a polynomial in ''<math>X</math>'' can be changed to a function in ''<math>x</math>'' by substitution. But the distinction is important because information may be lost when this substitution is made. For example, when working in [[Modular arithmetic|modulo 2]], we have that:

:<math>0 - 0^2 = 0, \quad 1 - 1^2 = 0,</math>

so the polynomial function ''<math>x - x^2</math>'' is identically equal to 0 for ''<math>x</math>'' having any value in the modulo-2 system. However, the polynomial ''<math>X - X^2</math>'' is not the zero polynomial, since the coefficients, 0, 1 and −1, respectively, are not all zero.