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How to do polynomial long division step by step?

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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.


Find
\frac{x^3 - 12x^2 - 42}{x-3}.
The problem is written like this:
\frac{x^3 - 12x^2 + 0x - 42}{x-3}.
The quotient and remainder can then be determined as follows:
  1. Divide the first term of the numerator by the highest term of the denominator. Place the result above the bar (x3 ÷ x = x3· x−1 = x3−1 =x2).
    \begin{matrix} x^2\\ \qquad\qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42} \end{matrix}







  2. 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 (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 numerator, and write the result underneath. This can be tricky at times, because of the sign. ((x3 − 12x2) − (x3 − 3x2) = −12x2 + 3x2 = −9x2) 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}







  4. 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}







  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, and the number left over (−123) is the remainder.
\frac{x^3 - 12x^2 - 42}{x-3} = \underbrace{x^2 - 9x - 27}_{q(x)} \underbrace{-\frac{123}{x-3}}_{r(x)/g(x)}
The long division algorithm for arithmetic can be viewed as a special case of the above algorithm.


Division transformation
Polynomial division allows for a polynomial to be written in a divisor–quotient form which is often advantageous. Consider polynomials P(x),D(x) where deg D < deg P. Then, for some quotient polynomial Q(x) and remainder polynomial R(x) with deg R < deg D,
\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \implies P(x) = D(x)Q(x) + R(x).
This rearrangement is known as the division transformation, and derives from the arithmetical identity {\mathrm{dividend} = \mathrm{divisor} \times \mathrm{quotient} + \mathrm{remainder} }. It has a variety of applications, including the determining of a remainder after polynomial division by a polynomial of degree ≥ 2.

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