Algebraic Fractions | 代数分式

📚 Algebraic Fractions | 代数分式

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions. Mastery of simplifying, operating, and solving equations with algebraic fractions is essential for A-Level Mathematics, as it underpins many topics including partial fractions, curve sketching, and calculus.

代数分式是分子、分母或两者都包含代数表达式的分式。掌握代数分式的简化、运算以及解含有代数分式的方程对于 A-Level 数学至关重要，因为它支撑着部分分式、曲线绘制和微积分等许多主题。

1. Simplifying Algebraic Fractions | 简化代数分式

To simplify an algebraic fraction, first factorise both numerator and denominator completely, then cancel any common factors. For example, simplify (x² – 4)/(x + 2). Factorise numerator as (x – 2)(x + 2). Cancel (x + 2) leaving x – 2, provided x ≠ –2.

要简化代数分式，首先将分子和分母完全因式分解，然后约去任何公因式。例如，简化 (x² – 4)/(x + 2)。将分子因式分解为 (x – 2)(x + 2)。约去 (x + 2) 得到 x – 2，前提是 x ≠ –2。

(x² – 4)/(x + 2) = x – 2, x ≠ –2

ariable to avoid division by zero. In the above, the original fraction is undefined at x = –2, so the simplified form is valid only for x ≠ –2.

始终要注明变量的限制条件以避免除以零。上述中，原分式在 x = –2 时无定义，因此简化形式仅在 x ≠ –2 时有效。

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2. Multiplying and Dividing Algebraic Fractions | 代数分式的乘除

merators together and denominators together, then simplify. For instance, (x/(x+1)) × ((x²–1)/x²). Factorise x²–1 = (x–1)(x+1).

要乘分式，将分子相乘、分母相乘，然后简化。例如，(x/(x+1)) × ((x²–1)/x²)。因式分解 x²–1 = (x–1)(x+1)。

So the product is (x (x–1)(x+1)) / ((x+1) x²). Cancel x+1 and one x, giving (x–1)/x, with x ≠ 0, –1.

因此乘积为 (x (x–1)(x+1)) / ((x+1) x²)。约去 x+1 和一个 x，得到 (x–1)/x，其中 x ≠ 0, –1。

For division, invert the second fraction and multiply. e.g., (2x/(x–3)) ÷ (4x²/(x²–9)). Factorise x²–9 = (x–3)(x+3). Then multiply by reciprocal: (2x/(x–3)) × ((x–3)(x+3)/(4x²)). Cancel (x–3) and x, resulting in (x+3)/(2x), x ≠ 0, 3, –3.

对于除法，将第二个分式取倒数再相乘。例如，(2x/(x–3)) ÷ (4x²/(x²–9))。因式分解 x²–9 = (x–3)(x+3)。然后乘以倒数：(2x/(x–3)) × ((x–3)(x+3)/(4x²))。约去 (x–3) 和 x，得到 (x+3)/(2x)，x ≠ 0, 3, –3。

(2x/(x–3)) ÷ (4x²/(x²–9)) = (x+3)/(2x), x ≠ 0, 3, –3

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3. Adding and Subtracting Algebraic Fractions | 代数分式的加减

same, simply add or subtract the numerators. Example: (2/(x+1)) + (x/(x+1)) = (2+x)/(x+1).

当分母相同时，直接将分子相加或相减。例如：(2/(x+1)) + (x/(x+1)) = (2+x)/(x+1)。

If denominators differ, find a common denominator, usually the lowest common multiple (LCM) of the denominators. Examples: 1/(x–2) + 3/(x+1). LCM is (x–2)(x+1).

如果分母不同，找到公分母，通常是分母的最小公倍数 (LCM)。例如：1/(x–2) + 3/(x+1)。LCM 是 (x–2)(x+1)。

Rewrite each fraction: (1(x+1) + 3(x–2)) / ((x–2)(x+1)) = (x+1 + 3x – 6)/((x–2)(x+1)) = (4x – 5)/((x–2)(x+1)).

重写每个分式：(1(x+1) + 3(x–2)) / ((x–2)(x+1)) = (x+1 + 3x – 6)/((x–2)(x+1)) = (4x – 5)/((x–2)(x+1))。

Always simplify the resulting numerator if possible. Do not cancel terms incorrectly; only factors can be cancelled.

如果可能，始终简化结果的分子。不要错误地约去项；只能约去因式。

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4. Finding the Lowest Common Denominator | 寻找最简公分母

The lowest common denominator (LCD) is the smallest expression that each denominator divides exactly. It is the LCM of the denominators. For numeric fractions, this is the least common multiple; for algebraic fractions, factorise each denominator and take each distinct factor raised to the highest power appearing.

最简公分母 (LCD) 是每个分母都能整除的最简表达式，也就是分母的最小公倍数。对于数值分数，这是最小公倍数；对于代数分式，将每个分母因式分解，取每个不同因式的最高次幂。

Example: denominators (x+2)², (x+2)(x–1), and (x–1). The LCD is (x+2)² (x–1).

例如，分母为 (x+2)², (x+2)(x–1), (x–1)。最简公分母是 (x+2)² (x–1)。

When adding or subtracting, adjust numerators accordingly by multiplying by the missing factors.

进行加减时，通过乘以所缺因式来调整分子。

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5. Equations Involving Algebraic Fractions | 包含代数分式的方程

To solve an equation with algebraic fractions, first find the common denominator and multiply every term by it to clear fractions. This yields a polynomial equation. Then solve using appropriate methods.

要解含有代数分式的方程，首先找到公分母，并将每一项乘以它来消去分母，从而得到一个多项式方程，然后用适当的方法求解。

Example: Solve 1/(x–2) + 3/(x+1) = 2. Multiply both sides by (x–2)(x+1): (x+1) + 3(x–2) = 2(x– get a quadratic, solve, and check for extraneous solutions that make any denominator zero.

例题：解 1/(x–2) + 3/(x+1) = 2。两边同乘 (x–2)(x+1) 得到：(x+1) + 3(x–2) = 2(x–2)(x+1)。展开并化简得到一个二次方程，求解并排除使任何分母为零的额外根。

Always verify solutions by substituting back into the original equation and ensuring denominators are non‑zero.

始终将解代回原方程验证，并确保分母不为零。

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6. Cross-Multiplication Technique | 交叉相乘法

When an equation has a single fraction on each side, cross‑multiplying can be efficient: if a/b = c/d, then ad = bc. This works if b and d are non‑zero.

当方程两边各有一个分式时，交叉相乘会很高效：如果 a/b = c/d，则 ad = bc。前提是 b 和 d 不为零。

Be careful: cross‑multiplying is only valid when the equation consists of exactly two equal fractions; do not apply it to sums or differences without first combining into a single fraction.

注意：交叉相乘仅在方程由两个相等的分式组成时才有效；不要直接用于和有加减的情况，除非先合并成一个分式。

Example: (x+2)/(x–3) = 4/(x). Cross‑multiply: x(x+2) = 4(x–3). Then solve the quadratic: x² + 2x = 4x – 12 → x² – 2x + 12 = 0 → no real roots.

例子：(x+2)/(x–3) = 4/(x)。交叉相乘：x(x+2) = 4(x–3)。然后

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