Leading coefficient

(noun)

The coefficient of the leading term.

Related Terms

  • Leading term

Examples of Leading coefficient in the following topics:

  • The Leading-Term Test

    • $a_nx^n$ is called the leading term of $f(x)$, while $a_n \not = 0$ is known as the leading coefficient.
    • The properties of the leading term and leading coefficient indicate whether $f(x)$ increases or decreases continually as the $x$-values approach positive and negative infinity:
    • which has $-\frac {x^4}{14}$ as its leading term and $- \frac{1}{14}$ as its leading coefficient.
    • 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.
  • The Remainder Theorem and Synthetic Division

    • Synthetic division only works for polynomials divided by linear expressions with a leading coefficient equal to $1.$
    • 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 $-27$:
    • Thus $1$ is a zero of a polynomial if and only if its coefficients add to $0.$
  • 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 $(x-x_0)^k$ equals $x_0^k$ for all positive integers $k$, we can substitute $x$ by $t+x_0$ 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 $x_0$.
    • In this case we substitute $x$ with $t+1$ and obtain a polynomial in $t$ with leading coefficient $3$ and constant term $1$.
  • The Discriminant

    • The discriminant of a polynomial is a function of its coefficients that reveals information about the polynomial's roots.
    • The discriminant of a quadratic function is a function of its coefficients that reveals information about its roots.
    • Where $a$, $b$, and $c$ are the coefficients in $f(x) = ax^2 + bx + c$.
    • Since adding zero and subtracting zero in the quadratic equation lead to the same outcome, there is only one distinct root of the quadratic function.
  • Factoring General Quadratics

    • In other words, the coefficient of the $x^2$ term is given by the product of the coefficients $\alpha_1$ and $\alpha_2$, and the coefficient of the $x$ term is given by the inner and outer parts of the FOIL process.
    • This leads to the factored form:
    • This leads to the equation:
  • 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 $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex.
    • If the coefficient $a>0$, the parabola opens upward, and if the coefficient $a<0$, the parabola opens downward.
    • The coefficients $b$ and $a$ together control the axis of symmetry of the parabola and the $x$-coordinate of the vertex.
    • The coefficient $c$ 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
  • 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, $x+x+x$ can be simplified as $3x$ (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 $a$ in a term $ax^by^c$ of the expanded version is known as a binomial coefficient.
    • Notice the coefficients are the numbers in row two of Pascal's triangle: $1,2,1$.
    • Where the coefficients $a_i$ in this expansion are precisely the numbers on row $n$ of Pascal's triangle.
    • Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of $y^n$ 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 $(x + y)^n$ into a sum involving terms of the form $ax^by^c$, where the exponents $b$ and $c$ are nonnegative integers with $b+c=n$, and the coefficient $a$ of each term is a specific positive integer depending on $n$ and $b$.
    • The coefficient a in the term of $ax^by^c$ is known as the binomial coefficient $n^b$ or $n^c$ (the two have the same value).
    • These coefficients for varying $n$ and $b$ can be arranged to form Pascal's triangle.
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