remainder

Examples of remainder in the following topics:

• The Remainder Theorem and Synthetic Division

  • Synthetic division is a technique for dividing a polynomial and finding the quotient and remainder.
  • It states that the remainder of a polynomial $f(x)$ divided by a linear divisor $(x-a)$ is equal to $f(a)$.
  • This gives the quotient $x^2-9x-27$ and the remainder $-123$.
  • To use the remainder theorem, one must first perform division, which is a bit of work.
  • Now we can also see what the remainder is, just by repeating the procedure:

• Dividing Polynomials

  • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.
  • The quotient and remainder can then be determined as follows: Divide the first term of the dividend by the highest term of the divisor (meaning the one with the highest power of $x$, which in this case is $x$): $x^3 \div x = x^2$.
  • The calculated polynomial is the quotient, and the number left over (−123) is the remainder: $x^3 - 12x^2 - 42 = (x - 3)(x^2 - 9x - 27) - 123$.
  • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.
  • The calculated polynomial is the quotient, and the number left over (−123) is the remainder:

• Division and Factors

  • Polynomial long division functions similarly to long division, and if the division leaves no remainder, then the divisor is called a factor.
  • So given two polynomials $D(x)$ (the dividend) and $d(x)$ (the divisor), we are looking for two polynomials $q(x)$ (the quotient) and $r(x)$ (the $remainder)$ such that $D(x) = d(x)q(x) + r(x)$ and the degree of $r(x)$ is strictly smaller than the degree of $d(x).$
  • Conceptually, we want to see how many copies of $d(x)$ are contained in $D(x)$ (this is the quotient) and then how far $D(x)$ is away from being a multiple of $d(x)$ (this is the remainder).
  • We see that the quotient $q(x)$ $3x^2+2x+6$ and the remainder $r(x)$ is $22$, so
  • If the remainder $r(x)$ equals $0$, we also say that there is no remainder and do not explicitly write out the $0$.

• Calcium and Phosphate Balance Regulation

  • Calcium is a key electrolyte: 99% is deposited in bone and the remainder is associated with hormone release and cell signalling.
  • Ninety-nine percent or more is deposited in bone and the remainder plays a vital role in nerve conduction, muscle contraction, hormone release and cell signalling.

• Residual Dividend Model

  • The Residual Dividend Model first uses earnings to finance new projects, then distributes the remainder as dividends.

• Linear Approximation

  • where $R_2$ is the remainder term (the difference between the actual value of $f(x)$ and the approximation found by the addition of the first two terms).

• Stereogenic Nitrogen

  • The non-identical mirror-image configurations are illustrated in the following diagram (the remainder of the molecule is represented by R, and the electron pair is colored yellow).

• Summary

  • In the remainder of the book, we will look at how social network analysts have formally translated some of the core concepts that social scientists use to describe social structures.

• Bilaminar Embryonic Disc Development

  • By the blastocyst stage, the embryo has become a hollow ball of cells with the inner cell mass (embryoblast) off to one side, while the blastocystic cavity completes the remainder of the sphere.

• The 22nd Amendment

  • But this article shall not apply to any person holding the office of President when this article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this article becomes operative from holding the office of President or acting as President during the remainder of such term.