Coefficient

A Coefficient is a multiplicative factor in a mathematical expression.

• Context:
  • It can range from being a Constant Coefficient to being a Variable Coefficient.
  • It can range from being a Leading Coefficient (with largest index) to being a Trailing Coefficient.
  • …
• Example(s):
  • the constants 6,4,1 in [math]\displaystyle{ 6x^3+4x+1 }[/math] (with 6 as leading coefficient).
  • The constants [math]\displaystyle{ a_i }[/math] in the polynomial [math]\displaystyle{ P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0 }[/math].
  • Binomial Coefficient,
  • Cartan Torsion Coefficient,
  • Clebsch-Gordan Coefficient,
  • Commutation Coefficient,
  • Correlation Coefficient,
  • Lagrangian Coefficient,
  • Multinomial Coefficient,
  • Pearson's Skewness Coefficient,
  • Quartile Variation Coefficient,
  • Racah V-Coefficient,
  • Racah W-Coefficient,
  • Regression Coefficient,
  • Roman Coefficient,
  • Triangle Coefficient,
  • Variation Coefficient.
  • …
• Counter-Example(s):
  • The coefficients of the basis vectors in the expression [math]\displaystyle{ v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n }[/math] can be variables and not necessary constants.
• See: Polynomial.

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References

2020

• (Mathworld, 2020) ⇒ Weisstein, Eric W. “Coefficient." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Coefficient.html
  • QUOTE: A multiplicative factor (usually indexed) such as one of the constants $a_i$ in the polynomial
    $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0$.
    .

    In this polynomial, the monomials are $x^n, x^{n-1}, \cdots, x$, and $1$, and the single variable is $x$.

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/coefficient Retrieved:2015-11-7.
  • In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but in any case does not involve any variables of the expression. For instance in : [math]\displaystyle{ 7x^2-3xy+1.5+y }[/math] the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem, as a, b, and c, where "c" is a constant, in : [math]\displaystyle{ ax^2+bx+c }[/math] when it is understood that these are not considered variables.

    Thus a polynomial in one variable x can be written as : [math]\displaystyle{ a_k x^k + \dotsb + a_1 x^1 + a_0 }[/math] for some integer [math]\displaystyle{ k }[/math] , where [math]\displaystyle{ a_k, \dotsc, a_1, a_0 }[/math] are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient.

    For the largest [math]\displaystyle{ i }[/math] with [math]\displaystyle{ a_i \ne 0 }[/math] (if any), [math]\displaystyle{ a_i }[/math] is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial : [math]\displaystyle{ \, 4x^5 + x^3 + 2x^2 }[/math] is 4.

    Specific coefficients arise in mathematical identities, such as the binomial theorem which involves binomial coefficients; these particular coefficients are tabulated in Pascal's triangle.

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