coefficient

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 $a$ 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

• 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 $-27$:
  • A special case of this is when the left number is $1$: then the last number equals the sum of all coefficients!
  • Thus $1$ is a zero of a polynomial if and only if its coefficients add to $0.$

• 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, $x+x+x$ 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.

• 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 $x^2 + 1$, 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 $(x-x_0)^k$ 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 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.
  • which has $-\frac {x^4}{14}$ as its leading term and $- \frac{1}{14}$ as its leading coefficient.
  • and the absolute value of $x$ is bigger than $MnK$, where $M$ is the absolute value of the largest coefficient divided by the leading coefficient, $n$ is the degree of the polynomial and $K$ is a big number, then the absolute value of $a_nx^n$ will be bigger than $nK$ times the absolute value of any other term, and bigger than $K$ 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 $A$ be the coefficient matrix, let $X$ be the variable matrix, and let $B$ 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.