多项式长除法

多项式长除法 是代数中的一种算法，用一个同次或低次的多项式去除另一个多项式。是常见算数技巧长除法的一个推广版本。它可以很容易地手算，因为它将一个相对复杂的除法问题分解成更小的一些问题。

例

计算

$\frac{x^3 - 12x^2 - 42}{x-3}.$

写成以下这种形式：

$\frac{x^3 - 12x^2 + 0x - 42}{x-3}.$

然后商和余数可以这样计算：

1. 将分子的第一项除以分母的最高次项（即次数最高的项，此处为x）。结果写在横线之上(x³ ÷ x = x²).

   $\begin{matrix} x^2\\ \qquad\qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42} \end{matrix}$

2. 将分母乘以刚得到结果（最终商的第一项），乘积写在分子前两项之下 (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. 从分子的相应项中减去刚得到的乘积（注意减一个负项相当于加一个正项），结果写在下面。((x³ − 12x²) − (x³ − 3x²) = −12x² + 3x² = −9x²)然后，将分子的下一项“拿下来”。

   $\begin{matrix} x^2\\ \qquad\qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42}\\ \qquad\;\; \underline{x^3 - 3x^2}\\ \qquad\qquad\qquad\quad\; -9x^2 + 0x \end{matrix}$

4. 重复前三步，只是现在用的是刚写作分子的那两项

   $\begin{matrix} \; x^2 - 9x\\ \qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42}\\ \;\; \underline{\;\;x^3 - \;\;3x^2}\\ \qquad\qquad\quad\; -9x^2 + 0x\\ \qquad\qquad\quad\; \underline{-9x^2 + 27x}\\ \qquad\qquad\qquad\qquad\qquad -27x - 42 \end{matrix}$

5. 重复第四步。这次没什么可以“拿下来”了。

   $\begin{matrix} \qquad\quad\;\, x^2 \; - 9x \quad - 27\\ \qquad\quad x-3\overline{) x^3 - 12x^2 + 0x - 42}\\ \;\; \underline{\;\;x^3 - \;\;3x^2}\\ \qquad\qquad\quad\; -9x^2 + 0x\\ \qquad\qquad\quad\; \underline{-9x^2 + 27x}\\ \qquad\qquad\qquad\qquad\qquad -27x - 42\\ \qquad\qquad\qquad\qquad\qquad \underline{-27x + 81}\\ \qquad\qquad\qquad\qquad\qquad\qquad\;\; -123 \end{matrix}$

横线之上的多项式即为商，而剩下的 (−123) 就是余数。

$\frac{x^3 - 12x^2 - 42}{x-3} = \underbrace{x^2 - 9x - 27}_{q(x)} \underbrace{-\frac{123}{x-3}}_{r(x)/g(x)}$

算数的长除法可以看做以上算法的一个特殊情形，即所有 x 被替换为10的情形。

除法变换

使用多项式长除法可以将一个多项式写成 除数-商 的形式（经常很有用）。 考虑多项式 P(x), D(x) （(D)的次数 < (P)的次数）。 然后，对某个商多项式 Q(x) 和余数多项式 R(x) （(R)的系数 < (D)的系数），

$\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \implies P(x) = D(x)Q(x) + R(x).$

这种变换叫做除法变换，是从算数等式 ${\mathrm{dividend} = \mathrm{divisor} \times \mathrm{quotient} + \mathrm{remainder} }$. 得到的。

应用

多项式的因式分解

Sometimes one or more roots of a polynomial are known, perhaps having been found using the rational root theorem. If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x - r)(Q(x)) where Q(x) is a polynomial of degree n–1. Q(x) is simply the quotient obtained from the division process; since r is known to be a root of P(x), it is known that the remainder must be zero.

Likewise, if more than one root is known, a linear factor (x – r) in one of them (r) can be divided out to obtain Q(x), and then a linear term in another root, s, can be divided out of Q(x), etc. Alternatively, they can all be divided out at once: for example the linear factors x– r and x – s can be multiplied together to obtain the quadratic factor x² – (r + s)x + rs, which can then be divided into the original polynomial Q(x) to obtain a quotient of degree n – 2.

In this way, sometimes all the roots of a polynomial of degree greater than four can be obtained, even though that is not always possible. For example, if the rational root theorem can be used to obtain a single (rational) root of a quintic polynomial, it can be factored out to obtain a quartic (fourth degree) quotient; the explicit formula for the roots of a quartic polynomial can then be used to find the other four roots of the quintic.

寻找多项式的切线

Polynomial long division can be used to find the equation of the line that is tangent to a polynomial at a particular point. If R(x) is the remainder when P(x) is divided by (x – r )² — that is, by x² – 2rx + r ² — then the equation of the tangent line to P(x) at x = r is y = R(x) (regardless of whether or not r is a root of the polynomial).

参见

• Polynomial remainder theorem
• Synthetic division, a more concise method of performing polynomial long division
• Ruffini's rule
• Euclidean domain
• Gröbner basis
• Greatest common divisor of two polynomials