Algebra 6: Algebraic Fractions | 代数6：代数分式

📚 Algebra 6: Algebraic Fractions | 代数6：代数分式

Algebraic fractions are a fundamental topic in IGCSE Edexcel Maths, bridging the gap between simple numeric fractions and more advanced algebra. They appear in simplification, equation solving and function manipulation, and a strong command of algebraic fractions is essential for achieving a top grade.

代数分式是 IGCSE Edexcel 数学的基础主题，连接了简单的数字分式与更高级的代数。它们出现在化简、解方程和函数处理中，扎实掌握代数分式对于取得高分至关重要。

1. What Are Algebraic Fractions? | 什么是代数分式？

An algebraic fraction has a numerator and/or a denominator that contains variables (like x or y) with operations. Examples include: (3x+2)/(x−1), 5/(x²+1) and (x²−9)/(2x+6). Just like numeric fractions, the denominator cannot be zero, so we often state restrictions such as x ≠ 1 in the first example.

代数分式的分子和/或分母中含有变量（如 x 或 y）及运算。例如：(3x+2)/(x−1)、5/(x²+1) 和 (x²−9)/(2x+6)。和数字分式一样，分母不能为零，因此我们通常会说明限制条件，比如第一个例子中 x ≠ 1。

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2. Simplifying Algebraic Fractions | 化简代数分式

To simplify, factorise both numerator and denominator completely, then cancel common factors. Always note any values that make the original denominator zero, as these are not part of the domain.

化简代数分式时，应先将分子和分母彻底因式分解，然后约去公因式。务必注明使原分母为零的取值，因为它们不在定义域内。

: x²−4 = (x−2)(x+2), x²−x−6 = (x−3)(x+2). Cancel (x+2) to get (x−2)/(x−3), with x ≠ −2, 3.

示例：化简 (x²−4)/(x²−x−6)。先因式分解：x²−4 = (x−2)(x+2)，x²−x−6 = (x−3)(x+2)。约去 (x+2)，得到 (x−2)/(x−3)，并且 x ≠ −2, 3。

(x²−4)/(x²−x−6) = (x−2)(x+2)/((x−3)(x+2)) = (x−2)/(x−3), x ≠ −2, 3

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3. Multiplying Algebraic Fractions | 代数分式的乘法

Multiply numerators together and denominators together. Always factorise and cancel before multiplying to keep numbers small. Cancel any factor from any numerator with any factor from any denominator.

将分子相乘、分母相乘。务必先因式分解并约分再乘，以保持数字简洁。任何分子中的因式都可以与任何分母中的因式约分。

(x+2) once: (x−1)(x+1)/(x+2) × (x+2)²/(x−1) = (x+1) × (x+2) = (x+1)(x+2), with x ≠ ±1, −2.

示例：化简 (x²−1)/(x+2) × (x+2)²/(x−1)。因式分解：x²−1 = (x−1)(x+1)。约去一次 (x+2)：原式 = (x+1)(x+2)，且 x ≠ ±1, −2。

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4. Dividing Algebraic Fractions | 代数分式的除法

To divide by an algebraic fraction, multiply by its reciprocal. Change the division sign to multiplication and flip the second fraction, then follow the multiplication rules.

除以一个代数分式时，乘以它的倒数。将除号换成乘号，翻转第二个分式，然后按乘法规则计算。

(x+3), 2x−4 = 2(x−2). Reciprocate: (x+3)/(x−2) × 2(x−2)/((x−3)(x+3)) = 2/(x−3), x ≠ 2, ±3.

示例：化简 (x+3)/(x−2) ÷ (x²−9)/(2x−4)。因式分解得 x²−9 = (x−3)(x+3)，2x−4 = 2(x−2)。取倒数相乘得 2/(x−3)，且 x ≠ 2, ±3。

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5. Adding and Subtracting with Same Denominator | 同分母的加减法

If two algebraic fractions have the identical denominator, simply add or subtract the numerators and keep the denominator. Then simplify if possible.

如果两个代数分式分母相同，只需将分子相加或相减，分母不变。然后尽可能化简。

Example: Simplify (3x+1)/(x+4) + (x−5)/(x+4) = (4x−4)/(x+4) = 4(x−1)/(x+4), with x ≠ −4.

示例：化简 (3x+1)/(x+4) + (x−5)/(x+4) = (4x−4)/(x+4) = 4(x−1)/(x+4)，且 x ≠ −4。

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6. Adding and Subtracting with Different Denominators | 异分母的加减法

When denominators differ, find the Lowest Common Denominator (LCD), usually the Lowest Common Multiple of the denominators. Convert each fraction to an equivalent fraction with the LCD, then add/subtract numerators and simplify.

当分母不同时，找到最小公分母（LCD），通常是分母的最小公倍数。将每个分式转换为以 LCD 为分母的等值分式，然后分子相加减并化简。

3)(x+2)) + 1(x−3)/((x−3)(x+2)) = (2x+4+x−3)/((x−3)(x+2)) = (3x+1)/((x−3)(x+2)), x ≠ 3, −2.

示例：化简 2/(x−3) + 1/(x+2)。最小公分母为 (x−3)(x+2)。通分：2(x+2)/((x−3)(x+2)) + (x−3)/((x−3)(x+2)) = (3x+1)/((x−3)(x+2))，且 x ≠ 3, −2。

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7. Solving Equations with Algebraic Fractions – Linear | 解含代数分式的线性方程

ions, multiply every term by the LCM of the denominators to clear fractions, then solve the resulting linear equation. Always check solutions against domain restrictions.

对于形如 a/b = c/d 的方程，交叉相乘得到 ad = bc。对于多个分式，将每一项乘以分母的最小公倍数以消去分母，然后解线性方程。始终对照定义域限制检验解。

3(x+1) = 2(x−1) → 3x+3 = 2x−2 → x = −5. Check denominators: x−1=−6≠0, x+1=−4≠0, so valid.

示例：解方程 3/(x−1) = 2/(x+1)。交叉相乘：3(x+1) = 2(x−1) → 3x+3 = 2x−2 → x = −5。检验分母：x−1=−6≠0，x+1=−4≠0，故解有效。

3/(x−1) = 2/(x+1) ⇒ 3(x+1) = 2(x−1) ⇒ x = −5

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8. Solving Equations Leading to Quadratics | 解引向二次方程的分式方程

Some algebraic fraction equations reduce to quadratics after multiplying by the LCD. Bring all terms to one side, factorise or use the quadratic formula, and then reject any root that makes a denominator zero.

有些代数分式方程在乘以 LCD 后会化为二次方程。将所有项移到一边，因式分解或使用求根公式，然后舍去使分母为零的根。

Example: Solve x/(x+2) + 1/(x−2) = 8/(x²−4). Note x²−4 = (

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