📚 Polynomial Long Division Explained | 多项式长除法详解
Polynomial long division is the algebraic version of arithmetic long division. It lets us divide one polynomial by another and write the result as a quotient and a remainder. This process is essential for solving higher-degree equations, sketching curves, simplifying algebraic fractions, and preparing for topics such as partial fractions.
多项式长除法是算术长除法的代数版本。它让我们可以用一个多项式去除另一个多项式，并把结果写成商式和余式的形式。这个过程在解高次方程、绘制函数图像、化简代数分式，以及为后续学习部分分式等内容做准备时都非常重要。
1. What Polynomial Long Division Achieves | 多项式长除法的目标
For any two polynomials P(x) and D(x), where D(x) is not the zero polynomial, we can perform division to obtain a quotient Q(x) and a remainder R(x). The division always produces the following relationship:
对任意两个多项式 P(x) 和 D(x)，其中 D(x) 不是零多项式，我们都可以做除法，得到商式 Q(x) 和余式 R(x)。除法运算始终满足如下关系：
P(x) = D(x) · Q(x) + R(x)
The remainder R(x) must have degree strictly less than the degree of the divisor D(x). If R(x) = 0, we say that D(x) divides P(x) exactly.
余式 R(x) 的次数必须严格小于除式 D(x) 的次数。如果 R(x) = 0，我们就称 D(x) 能整除 P(x)。
2. Setting Up the Division | 排列除式与被除式
Before starting the long division, write both the dividend and the divisor in descending powers of x. For example, 3 + 5x – x² + x³ should be rewritten as x³ – x² + 5x + 3.
开始长除法之前，应先把被除式和除式都按 x 的降幂排列。例如，3 + 5x – x² + x³ 应当改写成 x³ – x² + 5x + 3。
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If a term is missing, insert 0xⁿ as a placeholder.
如果缺少某一项，就用 0xⁿ 作为占位项。
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Placeholders keep the columns aligned during subtraction.
占位项可以保证在减法过程中各列对齐。
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Never skip a degree between the highest term and the constant term.
从最高次项到常数项之间，不要跳过一次项数。
3. The Step-by-Step Method | 长除法的逐步运算
The procedure is repeated until the degree of the remaining expression is smaller than the degree of the divisor.
重复以下步骤，直到剩余表达式的次数小于除式的次数为止。
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Step 1: Divide the first term of the dividend by the first term of the divisor.
第 1 步：用被除式的第一项除以除式的第一项。
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Step 2: Multiply the whole divisor by this result.
第 2 步：用这个结果去乘整个除式。
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Step 3: Subtract the product from the current dividend.
第 3 步：把乘积从当前被除式中减去。
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Step 4: Bring down the next term.
第 4 步：把下一项移下来。
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Step 5: Repeat from Step 1 until the remainder has a lower degree than the divisor.
第 5 步：从第 1 步开始重复，直到余式的次数低于除式的次数。
4. Worked Example 1: Exact Division | 示例一：整除
Divide x² + 5x + 6 by x + 2.
用 x + 2 去除 x² + 5x + 6。
(x² + 5x + 6) ÷ (x + 2) = x + 3, remainder 0
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First term: x² ÷ x = x.
第一项：x² ÷ x = x。
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Multiply: x(x + 2) = x² + 2x.
乘法：x(x + 2) = x² + 2x。
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Subtract: (x² + 5x + 6) – (x² + 2x) = 3x + 6.
相减：(x² + 5x + 6) – (x² + 2x) = 3x + 6。
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Next term: 3x ÷ x = 3.
下一项：3x ÷ x = 3。
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Multiply: 3(x + 2) = 3x + 6.
乘法：3(x + 2) = 3x + 6。
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Subtract: (3x + 6) – (3x + 6) = 0.
相减：(3x + 6) – (3x + 6) = 0。
The remainder is zero, so x + 2 is a factor, and x² + 5x + 6 = (x + 2)(x + 3).
余式为零，说明 x + 2 是它的因式，并且 x² + 5x + 6 = (x + 2)(x + 3)。
5. Worked Example 2: Division With a Remainder | 示例二：带余除法
Divide 2x³ – 5x² – x + 3 by x – 2.
用 x – 2 去除 2x³ – 5x² – x + 3。
(2x³ – 5x² – x + 3) ÷ (x – 2) = 2x² – x – 3, remainder -3
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First term: 2x³ ÷ x = 2x².
第一项：2x³ ÷ x = 2x²。
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Multiply: 2x²(x – 2) = 2x³ – 4x².
乘法：2x²(x – 2) = 2x³ – 4x²。
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Subtract: (2x³ – 5x² – x + 3) – (2x³ – 4x²) = -x² – x + 3.
相减：(2x³ – 5x² – x + 3) – (2x³ – 4x²) = -x² – x + 3。
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Next term: -x² ÷ x = -x.
下一项：-x² ÷ x = -x。
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Multiply: -x(x – 2) = -x² + 2x.
乘法：-x(x – 2) = -x² + 2x。
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Subtract: (-x² – x + 3) – (-x² + 2x) = -3x + 3.
相减：(-x² – x + 3) – (-x² + 2x) = -3x + 3。
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Next term: -3x ÷ x = -3.
下一项：-3x ÷ x = -3。
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Multiply: -3(x – 2) = -3x + 6.
乘法：-3(x – 2) = -3x + 6。
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Subtract: (-3x + 3) – (-3x + 6) = -3.
相减：(-3x + 3) – (-3x + 6) = -3。
Since the remainder -3 has degree zero, which is less than the degree of x – 2, the division stops. We can check the answer:
因为余式 -3 的次数为零，小于 x – 2 的次数，所以除法停止。我们可以验证答案：
(x – 2)(2x² – x – 3) – 3 = 2x³ – 5x² – x + 3
6. Using Placeholders for Missing Terms | 缺项时的零占位
When a polynomial has missing terms, the long division becomes much easier if we insert zero coefficients. For example, x³ + 1 can be written as x³ + 0x² + 0x + 1.
当多项式中缺少某些项时，插入零系数会让长除法容易得多。例如，x³ + 1 可以写成 x³ + 0x² + 0x + 1。
Divide x³ + 1 by x + 1:
用 x + 1 去除 x³ + 1：
(x³ + 1) ÷ (x + 1) = x² – x + 1, remainder 0
Without the 0x² and 0x terms, it is very easy to subtract the wrong terms. Placeholders also help when verifying that the quotient is correct.
如果没有 0x² 和 0x 这样的项，就很容易在相减时找错项。零占位项同时也有助于验证商式的正确性。
7. Remainder Theorem and Factor Theorem | 余式定理与因式定理
Polynomial long division is closely connected to two important theorems.
多项式长除法与以下两个重要定理紧密相关。
The remainder theorem states that when a polynomial P(x) is divided by x – a, the remainder is P(a).
余式定理指出：当多项式 P(x) 除以 x – a 时，余式等于 P(a)。
In the previous example, P(x) = 2x³ – 5x² – x + 3 and a = 2, so P(2) = 16 – 20 – 2 + 3 = -3. This matches the remainder we found by long division.
在前面的例子中，P(x) = 2x³ – 5x² – x + 3，且 a = 2，所以 P(2) = 16 – 20 – 2 + 3 = -3。这和我们用长除法得到的余式完全一致。
The factor theorem follows directly: if P(a) = 0, then x – a is a factor of P(x).
因式定理由此直接得出：如果 P(a) = 0，那么 x – a 是 P(x) 的一个因式。
8. Synthetic Division: A Faster Alternative | 综合除法：一种更快的方法
Synthetic division is a compressed form of polynomial long division. It works only when the divisor is linear, typically x – a.
综合除法是多项式长除法的一种简化形式。它只适用于除式为一次因式的情况，通常是 x – a。
Use the coefficients 2, -5, -1, 3 for P(x) = 2x³ – 5x² – x + 3 with a = 2:
取 P(x) = 2x³ – 5x² – x + 3 的系数 2、-5、-1、3，并令 a = 2：
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Bring down 2.
把 2 直接移下来。
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Multiply 2 by 2 to get 4; add -5 + 4 = -1.
2 乘以 2 得 4；-5 + 4 = -1。
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Multiply -1 by 2 to get -2; add -1 – 2 = -3.
-1 乘以 2 得 -2；-1 – 2 = -3。
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Multiply -3 by 2 to get -6; add 3 – 6 = -3.
-3 乘以 2 得 -6；3 – 6 = -3。
The last number, -3, is the remainder. The other numbers, 2, -1, -3, are the coefficients of the quotient 2x² – x – 3.
最后一个数 -3 是余式。其余的数 2、-1、-3 是商式 2x² – x – 3 的系数。
9. Common Mistakes to Avoid | 常见错误与易错点
Many students make the same errors when performing polynomial long division.
很多学生在做多项式长除法时都会犯类似的错误。
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Subtracting the entire product: when subtracting, change every sign of the product first.
忘记减去整个乘积：相减时要先改变乘积中每一项的符号。
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Forgetting placeholders for missing terms.
忘记为缺项添加零占位。
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Stopping too early: continue until the degree of the remainder is less than the degree of the divisor.
过早停止：应继续计算，直到余式的次数低于除式的次数。
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Mixing up the sign in synthetic division: dividing by x – 2 means a = 2, while dividing by x + 2 means a = -2.
综合除法中弄错符号：除以 x – 2 时 a = 2，而除以 x + 2 时 a = -2。
10. Practice Problems | 练习题
Try these questions yourself before checking the answers.
请先自己尝试完成以下问题，再对照答案。
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