Polynomial Division Techniques | 多项式除法运算技巧

📚 Polynomial Division Techniques | 多项式除法运算技巧

Polynomial division is one of the most frequently tested algebraic skills in A-Level Pure Mathematics. It appears not only in factorisation problems but also in curve sketching, integration, and partial fraction questions. Mastering this technique saves you valuable exam time and prevents costly errors.

多项式除法是 A-Level 纯数学中最高频考察的代数技巧之一。它不仅出现在因式分解问题中，还贯穿于曲线作图、积分和部分分式等题型。掌握这一技巧能为你在考场上节省宝贵时间，并避免代价高昂的失误。

1. Understanding Polynomial Division | 理解多项式除法

When we divide a polynomial f(x) by another polynomial d(x), we obtain a quotient q(x) and a remainder r(x). The fundamental relationship is: f(x) = d(x) × q(x) + r(x), where the degree of r(x) is always less than the degree of d(x).

当我们用一个多项式 f(x) 除以另一个多项式 d(x) 时，我们得到商式 q(x) 和余式 r(x)。基本关系为：f(x) = d(x) × q(x) + r(x)，其中 r(x) 的次数总是低于 d(x) 的次数。

For example, if we divide 17 by 5, we write 17 = 5 × 3 + 2. Similarly, when dividing polynomials, the remainder must have a smaller degree than the divisor. If dividing by a linear factor (x − a), the remainder is a constant.

例如，用 5 除 17，我们写成 17 = 5 × 3 + 2。同理，做多项式除法时，余式的次数必须低于除式的次数。如果用一次因式 (x − a) 作除式，则余式是一个常数。

f(x) = d(x) × q(x) + r(x)

2. Long Division: Step-by-Step | 长除法：分步详解

Long division is the most reliable method. Consider dividing f(x) = x³ − 6x² + 11x − 6 by x − 2.

长除法是最可靠的方法。我们以 f(x) = x³ − 6x² + 11x − 6 除以 x − 2 为例。

Step 1: Divide the leading term x³ by x to obtain x². Write x² as the first term of the quotient.

步骤一：用首项 x³ 除以 x 得到 x²，将 x² 作为商式的第一项。

Step 2: Multiply the divisor by x²: x²(x − 2) = x³ − 2x². Subtract this from the dividend: (x³ − 6x² + 11x − 6) − (x³ − 2x²) = −4x² + 11x − 6.

步骤二：将除式乘以 x²：x²(x − 2) = x³ − 2x²。从被除式中减去：(x³ − 6x² + 11x − 6) − (x³ − 2x²) = −4x² + 11x − 6。

Step 3: Repeat: −4x² ÷ x = −4x. Multiply: −4x(x − 2) = −4x² + 8x. Subtract: (−4x² + 11x − 6) − (−4x² + 8x) = 3x − 6.

步骤三：重复：−4x² ÷ x = −4x。相乘：−4x(x − 2) = −4x² + 8x。相减：(

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