📚 Polynomial Long Division | 多项式长除法
Polynomial long division is a fundamental algebraic technique that mirrors the long division of numbers, allowing us to divide one polynomial by another of equal or lower degree. It is essential for simplifying rational expressions, finding slant asymptotes, and applying the Remainder and Factor Theorems. Mastering this method builds a bridge between arithmetic and higher-level algebra, making it a core skill in the IB Mathematics: Analysis and Approaches and Applications and Interpretation syllabi.
多项式长除法是一种模仿数值长除法的基本代数技巧，可以将一个多项式除以另一个次数相等或更低的多项式。它对于化简有理式、寻找斜渐近线以及应用余数定理和因式定理至关重要。掌握这一方法在算术与高阶代数之间架起了桥梁，是 IB 数学分析与方法和应用与解释课程的核心技能。
1. Introduction to Polynomial Long Division | 多项式长除法简介
Long division for polynomials works on the same principle as dividing integers: we repeatedly determine how many times the leading term of the divisor 'fits' into the leading term of the current dividend, multiply, subtract, and bring down the next term. The process continues until the degree of the remainder is strictly less than the degree of the divisor.
多项式的长除法与整数除法的原理相同：我们不断确定除式的首项在被除式当前首项中“容纳”了多少次，然后相乘、相减并移下下一项。这一过程一直持续到余式的次数严格小于除式的次数为止。
ivisor, Q(x) is the quotient, and R(x) is the remainder, with deg(R) < deg(D) or R(x) = 0.
一个典型的除法等式可以写为 P(x) = D(x) * Q(x) + R(x)，其中 P(x) 是被除式，D(x) 是除式，Q(x) 是商式，R(x) 是余式，并且满足 deg(R) < deg(D) 或 R(x) = 0。
2. The Division Algorithm for Polynomials | 多项式的除法算法
x) * Q(x) + R(x), where the degree of R(x) is less than the degree of D(x). This theorem guarantees that long division always yields a unique quotient and remainder.
除法算法指出，给定两个多项式 P(x) 和 D(x)（D(x) ≠ 0），存在唯一的多项式 Q(x) 和 R(x)，使得 P(x) = D(x) * Q(x) + R(x)，且 R(x) 的次数小于 D(x) 的次数。该定理保证了长除法总能得出唯一的商式和余式。
If the remainder R(x) is zero, D(x) is a factor of P(x). This connects directly to the Factor Theorem, which we will explore later. The algorithm also underpins partial fraction decomposition and polynomial equation solving.
如果余式 R(x) 为零，则 D(x) 是 P(x) 的一个因式。这与我们稍后将探讨的因式定理直接相关。该算法也是部分分式分解和多项式方程求解的基础。
3. Step-by-Step Procedure | 逐步操作步骤
To divide P(x) by D(x), first arrange both polynomials in descending powers of x. If any powers are missing, insert them with a coefficient of 0. Then follow these steps: (1) Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient; (2) Multiply the entire divisor by this term; (3) Subtract the result from the current dividend; (4) Bring down the next term; (5) Repeat until the degree of the remainder is less than the degree of the divisor.
要用 D(x) 除以 P(x)，首先将两个多项式按 x 的降幂排列。如果缺少某些次幂，插入系数为 0 的项。然后按以下步骤操作：(1) 将被除式的首项除以除式的首项，得到商式的第一项；(2) 将整个除式乘以该项；(3) 从当前被除式中减去所得结果；(4) 移下下一项；(5) 重复此过程，直到余式的次数小于除式的次数。
It is crucial to keep terms aligned in columns according to their degree. Using a grid or careful alignment helps prevent errors, especially when subtracting polynomials with negative coefficients.
保持各项按次数对齐至关重要。使用网格或仔细对齐有助于避免错误，尤其是在减去带有负系数的多项式时。
4. Worked Example 1: Simple Division | 实例1：简单除法
Divide 2x³ + 3x² - x + 5 by x - 2.
计算 (2x³ + 3x² - x + 5) ÷ (x - 2)。
Step 1: Divide the leading term 2x³ by x to get 2x². Write 2x² above the x² column.
步骤1：将首项 2x³ 除以 x 得 2x²，写在 x² 列的上方。
from the dividend: (2x³ + 3x²) - (2x³ - 4x²) = 7x². Bring down the -x term to get 7x² - x.
步骤2：将 (x - 2) 乘以 2x² 得 2x³ - 4x²。从被除式中减去：(2x³ + 3x²) - (2x³ - 4x²) = 7x²。移下 -x 项，得到 7x² - x。
o get 7x² - 14x. Subtract: (7x² - x) - (7x² - 14x) = 13x. Bring down the 5 to get 13x + 5.
步骤3：将 7x² 除以 x 得 7x。将 (x - 2) 乘以 7x 得 7x² - 14x。相减：(7x² - x) - (7x² - 14x) = 13x。移下 5，得到 13x + 5。
(x - 2) by 13 to get 13x - 26. Subtract: (13x + 5) - (13x - 26) = 31. The remainder is 31.
步骤4：将 13x 除以 x 得 13。将 (x - 2) 乘以 13 得 13x - 26。相减：(13x + 5) - (13x - 26) = 31。余数为 31。
Quotient: 2x² + 7x + 13, Remainder: 31
商式：2x² + 7x + 13，余数：31
Thus, 2x³ + 3x² - x + 5 = (x - 2)(2x² + 7x + 13) + 31. Check by expanding.
因此，2x³ + 3x² - x + 5 = (x - 2)(2x² + 7x + 13) + 31。可通过展开验证。
5. Worked Example 2: Division with a Remainder | 实例2：带余数的除法
Divide 4x³ - 8x² + 2x - 1 by 2x + 1.
计算 (4x³ - 8x² + 2x - 1) ÷ (2x + 1)。
2x²(2x + 1) = 4x³ + 2x². Subtract: (-8x² - 2x²) = -10x². Bring down 2x to get -10x² + 2x.
首项 4x³ 除以 2x 得 2x²。乘除式：2x²(2x + 1) = 4x³ + 2x²。相减：(-8x² - 2x²) = -10x²。移下 2x 得到 -10x² + 2x。
tiply: -5x(2x + 1) = -10x² - 5x. Subtract: (2x - (-5x)) = 7x. Bring down -1 to get 7x - 1.
-10x² 除以 2x 得 -5x。乘：-5x(2x + 1) = -10x² - 5x。相减：(2x - (-5x)) = 7x。移下 -1 得 7x - 1。
2. Multiply: (7/2)(2x + 1) = 7x + 7/2. Subtract: (-1 - 7/2) = -9/2. The remainder is -9/2.
7x 除以 2x 得 7/2。乘：(7/2)(2x + 1) = 7x + 7/2。相减：(-1 - 7/2) = -9/2。余数为 -9/2。
Quotient: 2x² - 5x + 7/2, Remainder: -9/2
商式：2x² - 5x + 7/2，余数：-9/2
Thus, 4x³ - 8x² + 2x - 1 = (2x + 1)(2x² - 5x + 7/2) - 9/2.
因此，4x³ - 8x² + 2x - 1 = (2x + 1)(2x² - 5x + 7/2) - 9/2。
6. Handling Missing Terms | 处理缺项
When the dividend has missing powers of x, such as x³ + 2x - 5, it is essential to insert 0x² as a placeholder. This keeps the column alignment correct and prevents misplacement of terms during subtraction. For example, rewrite x³ + 2x - 5 as x³ + 0x² + 2x - 5 before dividing by x - 1.
当被除式缺少 x 的某些次幂时（例如 x³ + 2x - 5），必须插入 0x² 作为占位项。这样可以保持列对齐，避免减法过程中各项错位。例如，在除以 x - 1 之前，先将 x³ + 2x - 5 重写为 x³ + 0x² + 2x - 5。
Failing to include placeholders often leads to an incorrect quotient or remainder, particularly when later terms are subtracted from zero coefficients. The same principle applies if the divisor itself has missing terms: for consistency, write both polynomials with all degrees represented.
不添加占位项往往会导致商式或余式错误，尤其是在从零系数中减去后续项时。如果除式本身也有缺项，同样适用这一原则：为保持一致性，将两个多项式都写成所有次数均出现的形式。
7. Relationship with Synthetic Division | 与综合除法的关系
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - c. It uses only the coefficients and the constant c, reducing the amount of writing and calculation. However, synthetic division only works when the divisor is linear with a leading coefficient of 1. For divisors like 2x + 1 or non-linear divisors, polynomial long division must be used.
综合除法是一种用于除以形如 x - c 的线性因式的快捷方法。它仅利用系数和常数 c，减少了书写和计算量。然而，综合除法仅适用于首项系数为 1 的线性除式。对于像 2x + 1 这样的除式或非线性除式，则必须使用多项式长除法。
| Aspect | Polynomial Long Division | Synthetic Division |
|---|---|---|
| Divisor form | Any polynomial D(x) | Only x - c |
| Method | Write all terms, multiply, subtract | Bring down, multiply by c, add |
| Risk of error | Alignment mistakes | Sign errors with c |
| 方面 | 多项式长除法 | 综合除法 |
|---|---|---|
| 除式形式 | 任意多项式 D(x) | 仅限 x - c |
| 方法 | 写出所有项，乘、减 | 移下，乘以 c，相加 |
| 易错点 | 对齐错误 | 与 c 相关的符号错误 |
Both methods yield the same quotient and remainder when applicable. For IB examinations, you are expected to be proficient in both, but long division is the more general and robust tool.
两种方法在适用时都会得出相同的商式和余式。IB 考试期望考生熟练掌握两者，但长除法是更通用、更稳健的工具。
8. The Remainder Theorem and Factor Theorem | 余数定理与因式定理
, P(2) = 2(8) + 3(4) - 2 + 5 = 16 + 12 - 2 + 5 = 31, matching the remainder found earlier.
余数定理指出，当多项式 P(x) 除以 (x - c) 时，余数等于 P(c)。这使得无需进行完整除法就能快速求出余数。例如，在实例1中，P(2) = 2(8) + 3(4) - 2 + 5 = 16 + 12 - 2 + 5 = 31，与之前求得的余数一致。
Polynomial long division can then be used to factorise P(x) completely once one factor is known, reducing it to a product of linear and irreducible quadratic factors.
因式定理由此得出：(x - c) 是 P(x) 的因式，当且仅当 P(c) = 0。一旦已知一个因式，就可以利用多项式长除法将 P(x) 完全分解为线性和不可约二次因式的乘积。
In an IB problem, you may be given one root and asked to find the others by dividing the polynomial by (x - root) and then solving the resulting quadratic equation.
在 IB 问题中，你可能会已知一个根，然后要求通过将多项式除以 (x - root) 并解所得的二次方程来找出其余根。
9. Applications: Factoring and Solving Equations | 应用：因式分解与解方程
n factorise the quadratic: x² - 5x + 6 = (x - 2)(x - 3). Hence the roots are x = -1, 2, 3.
考虑三次方程 x³ - 4x² + x + 6 = 0。通过试值，x = -1 得 (-1)³ - 4(-1)² - 1 + 6 = 0，因此 (x + 1) 是一个因式。用长除法将 x³ - 4x² + x + 6 除以 x + 1，商式为 x² - 5x + 6，余数为 0。然后分解二次式：x² - 5x + 6 = (x - 2)(x - 3)。从而根为 x = -1, 2, 3。
This method of factorising by finding one rational root via the Rational Root Theorem and then performing long division is a standard technique for solving higher-degree polynomial equations in IB Mathematics.
这种通过有理根定理找到一个有理根，然后进行长除法的因式分解方法，是 IB 数学中求解高次多项式方程的标准技巧。
10. Graphical Interpretation: Asymptotes | 图形解释：渐近线
remainder 6. So f(x) = x + 4 + 6/(x - 1). As x → ±∞, 6/(x - 1) → 0, hence the slant asymptote is y = x + 4.
多项式长除法可用于求有理函数的斜渐近线方程，此时分子的次数恰好比分母高一次。对于 f(x) = (x² + 3x + 2) / (x - 1)，进行长除法：x² + 3x + 2 除以 x - 1 得商式 x + 4，余数 6。所以 f(x) = x + 4 + 6/(x - 1)。当 x → ±∞ 时，6/(x - 1) → 0，因此斜渐近线为 y = x + 4。
If the remainder is zero, the rational function simplifies to a polynomial (a whole function), meaning the graph is just the polynomial curve without any vertical asymptote from that factor. Long division reveals the 'whole part' of a rational expression, which is highly useful in curve sketching.
如果余数为零，有理函数简化为多项式（整函数），意味着图像就是该多项式曲线，没有由该因式产生的垂直渐近线。长除法揭示了有理式的“整式部分”，这在绘制曲线草图时非常有用。
11. Common Mistakes to Avoid | 常见错误
One frequent error is forgetting to distribute the subtraction sign when subtracting the product of the divisor and a quotient term. Always subtract the entire expression: P(x) - (D(x) * term), which changes the signs of every term in the product. Another is misaligning terms of different degrees; using graph paper or carefully drawn columns helps.
一个常见的错误是在减去除式与商式项的乘积时忘记分配减号。一定要减去整个表达式：P(x) - (D(x) * term)，这会改变乘积中每一项的符号。另一个错误是不同次数的项没有对齐；使用方格纸或仔细划列有助于避免此问题。
Students sometimes stop dividing too early, leaving a remainder whose degree is equal to or greater than the divisor's degree. The process must continue until the remainder is of lower degree. Additionally, when encountering fractional coefficients, maintain them as exact fractions rather than decimal approximations to preserve precision.
学生有时会过早停止除法，留下次数等于或大于除式次数的余式。必须继续下去，直到余式的次数更低。此外，当遇到分数系数时，应保持精确分数形式而非小数近似，以保证精确性。
12. Practice Tips and Summary | 练习技巧与总结
x) * Q(x) + R(x) holds true. After division, expand the right-hand side and confirm it matches the original dividend.
要熟练掌握多项式长除法，应练习多种除式：线性的、二次的以及缺项的。始终通过验证 P(x) = D(x) * Q(x) + R(x) 是否成立来检查结果。除法完成后，展开右侧并确认是否与原被除式一致。
Remember that this technique is not isolated; it connects to the Remainder and Factor Theorems, synthetic division, partial fractions, and asymptotic analysis. Building a solid foundation in long division will support success in calculus topics like integration of rational functions and in solving polynomial inequalities.
请记住，这一技巧并非孤立存在；它与余数定理、因式定理、综合除法、部分分式以及渐近分析相联系。在长除法上打下坚实基础，将有助于在有理函数积分等微积分专题以及求解多项式不等式中取得成功。
Polynomial long division: an essential tool that transforms complex rational expressions into manageable forms.
多项式长除法：一项将复杂有理式转化为易处理形式的关键工具。
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