Polynomial Long Division - Example

Example

Find the quotient and the remainder of the division of

the dividend by

the divisor.

The dividend is first rewritten like this:

The quotient and remainder can then be determined as follows:

  1. 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). Place the result above the bar (x3 ÷ x = x2).
    \begin{matrix} x^2\\ \qquad\qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42} \end{matrix}
  2. Multiply the divisor by the result just obtained (the first term of the eventual quotient). Write the result under the first two terms of the numerator (x2 · (x − 3) = x3 − 3x2).
    \begin{matrix} x^2\\ \qquad\qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42}\\ \qquad\;\; x^3 - 3x^2 \end{matrix}
  3. Subtract the product just obtained from the appropriate terms of the original dividend (being careful that subtracting something having a minus sign is equivalent to adding something having a plus sign), and write the result underneath ((x3 − 12x2) − (x3 − 3x2) = −12x2 + 3x2 = −9x2) Then, "bring down" the next term from the dividend.
    \begin{matrix} x^2\\ \qquad\qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42}\\ \qquad\;\; \underline{x^3 - 3x^2}\\ \qquad\qquad\qquad\quad\; -9x^2 + 0x \end{matrix}
  4. Repeat the previous three steps, except this time use the two terms that have just been written as the dividend.
    \begin{matrix} \; x^2 - 9x\\ \qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42}\\ \;\; \underline{\;\;x^3 - \;\;3x^2}\\ \qquad\qquad\quad\; -9x^2 + 0x\\ \qquad\qquad\quad\; \underline{-9x^2 + 27x}\\ \qquad\qquad\qquad\qquad\qquad -27x - 42 \end{matrix}
  5. Repeat step 4. This time, there is nothing to "pull down".
    \begin{matrix} \qquad\quad\;\, x^2 \; - 9x \quad - 27\\ \qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42}\\ \;\; \underline{\;\;x^3 - \;\;3x^2}\\ \qquad\qquad\quad\; -9x^2 + 0x\\ \qquad\qquad\quad\; \underline{-9x^2 + 27x}\\ \qquad\qquad\qquad\qquad\qquad -27x - 42\\ \qquad\qquad\qquad\qquad\qquad \underline{-27x + 81}\\ \qquad\qquad\qquad\qquad\qquad\qquad\;\; -123 \end{matrix}

The polynomial above the bar is the quotient q(x), and the number left over (−123) is the remainder r(x).

The long division algorithm for arithmetic is very similar to the above algorithm, in which the variable x is replaced by the specific number 10.

Read more about this topic:  Polynomial Long Division

Famous quotes containing the word example:

    “Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
    —Marcel Proust (1871–1922)