Editing Coefficient (section)

== Terminology and definition ==
In mathematics, a '''coefficient''' is a multiplicative factor in some [[Summand|term]] of a [[polynomial]], a [[series (mathematics)|series]], or any [[expression (mathematics)|expression]].  For example, in the polynomial
<math display="block">7x^2-3xy+1.5+y,</math>
with variables <math>x</math> and <math>y</math>, the first two terms have the coefficients 7 and −3. The third term 1.5 is the constant coefficient. In the final term, the coefficient is 1 and is not explicitly written.

In many scenarios, coefficients are numbers (as is the case for each term of the previous example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish between symbols representing variables and symbols representing parameters. Following [[René Descartes]], the variables are often denoted by {{mvar|x}}, {{mvar|y}}, ..., and the parameters by {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, ..., but this is not always the case. For example, if {{mvar|y}} is considered a parameter in the above expression, then the coefficient of {{mvar|x}} would be {{math|−3''y''}}, and the constant coefficient (with respect to {{mvar|x}}) would be {{math|1.5 + ''y''}}.

When one writes 
<math display="block">ax^2+bx+c,</math>
it is generally assumed that {{mvar|x}} is the only variable, and that {{mvar|a}}, {{mvar|b}} and {{mvar|c}} are parameters; thus the constant coefficient is {{mvar|c}} in this case.

{{anchor|leading coefficient}}Any [[polynomial]] in a single variable {{mvar|x}} can be written as
<math display="block">a_k x^k + \dotsb + a_1 x^1 + a_0</math>
for some [[nonnegative integer]] <math>k</math>, where <math>a_k, \dotsc, a_1, a_0</math> are the coefficients.  This includes the possibility that some terms have coefficient 0; for example, in <math>x^3 - 2x + 1</math>, the coefficient of <math>x^2</math> is 0, and the term <math>0x^2</math> does not appear explicitly.  For the largest <math>i</math> such that <math>a_i \ne 0</math> (if any), <math>a_i</math> is called the '''leading coefficient''' of the polynomial. For example, the leading coefficient of the polynomial
<math display="block">4x^5 + x^3 + 2x^2</math>
is 4. This can be generalised to multivariate polynomials with respect to a [[monomial order]], see {{section link|Gröbner basis|Leading term, coefficient and monomial}}.