In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised 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.
Example
Find
- \frac{x^3 - 12x^2 - 42}{x-3}.
The problem is written like this:
- x-3\overline{) x^3 - 12x^2 0x - 42}.
The quotient and remainder can then be determined as follows:
- Divide the first term of the numerator by the highest term of the denominator. Place the result above the bar (x 3 ÷ x = x 3 · x −1 = x 3 −1 = x 2 ).
\begin
... see moreIn algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised 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.
Example
Find
- \frac{x^3 - 12x^2 - 42}{x-3}.
The problem is written like this:
- x-3\overline{) x^3 - 12x^2 0x - 42}.
The quotient and remainder can then be determined as follows:
- Divide the first term of the numerator by the highest term of the denominator. Place the result above the bar (x 3 ÷ x = x 3 · x −1 = x 3 −1 = x 2 ).
\begin{matrix} x^2\\ \qquad\qquad\quad x-3\overline{) x^3 - 12x^2 0x - 42} \end{matrix}
- Multiply the denominator by the result just obtained (the first term of the eventual quotient). Write the result under the first two terms of the numerator (x 2 · (x − 3) = x 3 − 3x 2 ).
\begin{matrix} x^2\\ \qquad\qquad\quad x-3\overline{) x^3 - 12x^2 0x - 42}\\ \qquad\;\; x^3 - 3x^2 \end{matrix}
- Subtract the product just obtained from the appropriate terms of the original numerator, and write the result underneath. This can be tricky at times, because of the sign. ((x 3 − 12x 2 ) − (x 3 − 3x 2 ) = −12x 2 3x 2 = −9x 2 ) Then, "bring down" the next term from the numerator.
\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}
- Repeat the previous three steps, except this time use the two terms that have just been written as the numerator.
\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}
- 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}
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