quotient

Examples of quotient in the following topics:

• Logarithms of Quotients

  • By applying the product, power, and quotient rules, you could write this expression as $\log_2(x^4)+\log_2(y^9)-\log_2(z^{100}) = 4\log_2x+9\log_2y-100\log_2z.$
  • Relate the quotient rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of quotients

• Difference Quotients

  • The difference quotient is used in algebra to calculate the average slope between two points but has broader effects in calculus.
  • It is also known as Newton's quotient:
  • The difference quotient is the average slope of a function between two points.
  • In this case, the difference quotient is know as a derivative, a useful tool in calculus.
  • Relate the difference quotient in algebra to the derivative in calculus

• 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.
  • Multiply the divisor by the result just obtained (the first term of the eventual quotient): $x^2 \cdot (x − 3) = x^3 − 3x^2$.
  • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.
  • Multiply the divisor by the result just obtained (the first term of the eventual quotient): $x^2 \cdot (x − 3) = x^3 − 3x^2$.
  • The calculated polynomial is the quotient, and the number left over (−123) is the remainder:

• The Remainder Theorem and Synthetic Division

  • Synthetic division is a technique for dividing a polynomial and finding the quotient and remainder.
  • This gives the quotient $x^2-9x-27$ and the remainder $-123$.
  • As the leading coefficient of the divisor is $1$, the leading coefficient of the quotient is the same as that of the dividend:
  • The result of $-12 + 3$ is $9$, so since the leading coefficient of the divisor is still $1$, the second coefficient of the quotient is $-9:$
  • So the quotient must be the second degree polynomial $x^2 + 9x + 27$.

• Division and Factors

  • 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).
  • Again looking at the highest degree terms, we see that $4x^2 = 2x\cdot2x$, so we write down $2x$ as the second term in the quotient and proceed as before:
  • We see that the quotient $q(x)$ $3x^2+2x+6$ and the remainder $r(x)$ is $22$, so
  • (Of course, the quotient will also be a factor.)

• Sums, Differences, Products, and Quotients

• Zeroes of Polynomial Functions With Rational Coefficients

  • In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
  • It was thus named in 1895 by Peano after quoziente, Italian for "quotient".

• Basic Operations

  • Rather than multiplying quantities together to result in a larger value, you are splitting a quantity into a smaller value, called the quotient.
  • Calculate the sum, difference, product, and quotient of positive whole numbers

• Negative Numbers

  • Calculate the sum, difference, product, and quotient of negative whole numbers

• Fitting a Curve

  • The denominator is $92-\frac{1}{8}(20)^{2}=92-50=42$ and the slope is the quotient of the numerator and denominator: $\frac{23.25}{42}\approx0.554.$