long division

Examples of long division in the following topics:

• Dividing Polynomials

  • Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree.
  • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.
  • Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree.This method is a generalized version of the familiar arithmetic technique called long division.It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
  • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.

• 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.
  • We want to find integers $q$ and $r$ such that $0 \leq r < d$ and $D = qd+r.$ This we can do with long division, which we all learned to do in elementary school.
  • The beauty of long division is that the algorithm can be used not for integers only, but also for polynomials.

• The Remainder Theorem and Synthetic Division

  • In algebra, the polynomial remainder theorem or little Bézout's theorem, is an application of polynomial long division.
  • An shorthand way to perform long division is synthetic division.
  • It also takes significantly less space than long division.
  • Most importantly, the subtractions in long division are converted to additions by switching the signs at the very beginning, preventing sign errors.
  • This is the same as long division, but it uses less lines and no variables.

• Integer Coefficients and the Rational Zeroes Theorem

  • Once we have found all the rational zeroes (and counted their multiplicity, for example by dividing using long division), we know that the number of irrational and complex roots.
  • These root candidates can be tested, either by plugging them in directly, or by dividing and checking to see whether there is any remainder, for example using long division.
  • Moreover, once we have established a root, we must use division anyway to check whether it is a multiple root.
  • The disadvantage is that we have to use long division more often.
  • When there are a lot of zero candidates for a small degree polynomial, we may just want to plug in candidates and only use division when we have found a root.

• Sums, Differences, Products, and Quotients

  • Dividing equations uses similar theory as multiplying, since division is the equivalent of multiplying by the inverse.
  • Use the idea of combining like terms to add and subtract functions, and the FOIL method and long division to multiply and divide functions

• Partial Fractions

  • It is necessary to perform the Euclidean division of $f$ by $g$ using polynomial long division, giving $f(x) = E(X)g(x) + h(x)$.

• Rules for Solving Inequalities

  • As long as the same value is added or subtracted from both sides, the resulting inequality remains true.
  • The properties that deal with multiplication and division state, for any real numbers, $a$, $b$ and non-zero $c$:
  • To see how the rules for multiplication and division apply, consider the inequality $2x > 8$.

• Complex Conjugates and Division

  • The complex conjugate of x + yi is x - yi, and the division of two complex numbers can be defined using the complex conjugate.
  • The division of two complex numbers is defined in terms of complex multiplication (described above) and real division.
  • Division can be defined in this way because of the following observation:
  • Neither the real part c nor the imaginary part d of the denominator can be equal to zero for division to be defined.

• Basic Operations

  • The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division.
  • Division is the inverse of multiplication.

• Division of Complex Numbers

  • Division of complex numbers is accomplished by multiplying by the multiplicative inverse.
  • The key is to think of division by a number $z$ as multiplying by the multiplicative inverse of $z$.