coefficient

(noun)

a constant by which an algebraic term is multiplied.

Related Terms

  • parameter
  • coefficients
  • term
  • parameters
  • variable
  • unknown
  • linear
  • degree
  • polynomial
  • trinomial

(noun)

A quantity (usually a number) that remains the same in value within a problem.

Related Terms

  • parameter
  • coefficients
  • term
  • parameters
  • variable
  • unknown
  • linear
  • degree
  • polynomial
  • trinomial

Examples of coefficient in the following topics:

  • Graphing Quadratic Equations In Standard Form

    • Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function's graph.
    • The coefficient aaa controls the speed of increase (or decrease) of the quadratic function from the vertex.
    • If the coefficient a>0a>0a>0, the parabola opens upward, and if the coefficient a<0a<0a<0, the parabola opens downward.
    • The coefficients bbb and aaa together control the axis of symmetry of the parabola and the xxx-coordinate of the vertex.
    • The coefficient ccc controls the height of the parabola.
  • Zeroes of Polynomial Functions With Rational Coefficients

    • Polynomials with rational coefficients should be treated and worked the same as other polynomials.
    • Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals".
    • However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.
    • Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations.
    • Extend the techniques of finding zeros to polynomials with rational coefficients
  • The Remainder Theorem and Synthetic Division

    • We start by writing down the coefficients from the dividend and the negative second coefficient of the divisor.
    • Bring down the first coefficient and multiply it by the divisor.
    • Then add the next column of coefficients, get the result and multiply that by the divisor to find the third coefficient −27-27−27:
    • A special case of this is when the left number is 111: then the last number equals the sum of all coefficients!
    • Thus 111 is a zero of a polynomial if and only if its coefficients add to 0.0.0.
  • Simplifying Algebraic Expressions

    • A coefficient is a numerical value which multiplies a variable (the operator is omitted).
    • When a coefficient is one, it is usually omitted.
    • Added terms are simplified using coefficients.
    • For example, x+x+xx+x+xx+x+x can be simplified as 3x3x3x (where 3 is the coefficient).
    • 1 – Exponent (power), 2 – Coefficient, 3 – term, 4 – operator, 5 – constant, x,y – variables
  • Binomial Expansions and Pascal's Triangle

    • The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial.
    • Any coefficient aaa in a term axbycax^by^cax​b​​y​c​​ of the expanded version is known as a binomial coefficient.
    • Notice the coefficients are the numbers in row two of Pascal's triangle: 1,2,11,2,11,2,1.
    • Where the coefficients aia_ia​i​​ in this expansion are precisely the numbers on row nnn of Pascal's triangle.
    • Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of yny^ny​n​​ in these binomial expansions, while the next diagonal corresponds to the coefficient of $xy^{n−1}$ and so on.
  • Total Number of Subsets

    • The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.
    • According to the theorem, it is possible to expand the power (x+y)n(x + y)^n(x+y)​n​​ into a sum involving terms of the form axbycax^by^cax​b​​y​c​​, where the exponents bbb and ccc are nonnegative integers with b+c=nb+c=nb+c=n, and the coefficient aaa of each term is a specific positive integer depending on nnn and bbb.
    • The coefficient a in the term of axbycax^by^cax​b​​y​c​​ is known as the binomial coefficient nbn^bn​b​​ or ncn^cn​c​​ (the two have the same value).
    • These coefficients for varying nnn and bbb can be arranged to form Pascal's triangle.
  • The Fundamental Theorem of Algebra

    • The fundamental theorem states that every non-constant, single-variable polynomial with complex coefficients has at least one complex root.
    • Some polynomials with real coefficients, like x2+1x^2 + 1x​2​​+1, have no real zeros.
    • As it turns out, every polynomial with a complex coefficient has a complex zero.
    • Every polynomial of odd degree with real coefficients has a real zero.
    • In particular, every polynomial of odd degree with real coefficients admits at least one real root
  • Integer Coefficients and the Rational Zeros Theorem

    • When given a polynomial with integer coefficients, we can plug in all of these candidates and see whether they are a zero of the given polynomial.
    • Since every polynomial with rational coefficients can be multiplied with an integer to become a polynomial with integer coefficients and the same zeros, the Rational Root Test can also be applied for polynomials with rational coefficients.
    • Now we use a little trick: since the constant term of (x−x0)k(x-x_0)^k(x−x​0​​)​k​​ equals x0kx_0^kx​0​k​​ for all positive integers kkk, we can substitute xxx by t+x0t+x_0t+x​0​​ to find a polynomial with the same leading coefficient as our original polynomial and a constant term equal to the value of the polynomial at x0x_0x​0​​.
    • In this case we substitute xxx with t+1t+1t+1 and obtain a polynomial in ttt with leading coefficient 333 and constant term 111.
  • The Leading-Term Test

    • anxna_nx^na​n​​x​n​​ is called the leading term of f(x)f(x)f(x), while $a_n \not = 0$ is known as the leading coefficient.
    • which has −x414-\frac {x^4}{14}−​14​​x​4​​​​ as its leading term and −114- \frac{1}{14}−​14​​1​​ as its leading coefficient.
    • and the absolute value of xxx is bigger than MnKMnKMnK, where MMM is the absolute value of the largest coefficient divided by the leading coefficient, nnn is the degree of the polynomial and KKK is a big number, then the absolute value of anxna_nx^na​n​​x​n​​ will be bigger than nKnKnK times the absolute value of any other term, and bigger than KKK times the other terms combined!
    • As the degree is even and the leading coefficient is negative, the function declines both to the left and to the right.
    • Because the degree is odd and the leading coefficient  is positive, the function declines to the left and inclines to the right.
  • Matrix Equations

    • Make sure that one side of the equation is only variables and their coefficients, and the other side is just constants.
    • To solve a system of linear equations using an inverse matrix, let AAA be the coefficient matrix, let XXX be the variable matrix, and let BBB be the constant matrix.
    • If the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be dependent and have infinitely many solutions.
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