remainder

(noun)

The amount left over after subtracting the divisor as many times as possible from the dividend without producing a negative result. If (dividend) and $d$ (divisor) are integers, then can always be expressed in the form $n = dq + r$, where $q$ (quotient) and $r$ (remainder) are also integers and $0 \leq r \leq d$ .

Related Terms

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 $0$.

  • Finding Zeros of Factored Polynomials

    • In general, we know from the remainder theorem that $a$ is a zero of $f(x)$ if and only if $x-a$ divides $f(x).$ Thus if we can factor $f(x)$ in polynomials of as small a degree as possible, we know its zeros by looking at all linear terms in the factorization.

  • Integer Coefficients and the Rational Zeros Theorem

    • 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.