Algebraic Fractions | 代数分式

📚 Algebraic Fractions | 代数分式

Algebraic fractions are expressions where the numerator or denominator (or both) contain polynomial terms. They appear throughout A‑Level Mathematics, from simplifying rational expressions and solving equations to partial fraction decomposition and curve sketching. A solid command of algebraic fractions underpins success in topics such as calculus, binomial expansion and proof.

代数分式是指分子或分母（或两者）中含有多项式项的表达式。它们贯穿整个 A‑Level 数学课程，从化简有理式、解方程，到部分分式分解和曲线草图绘制。扎实掌握代数分式是学好微积分、二项式展开和证明等主题的基础。

1. What is an Algebraic Fraction? | 什么是代数分式？

An algebraic fraction is a fraction whose numerator and denominator are polynomials. For example, (2x+1)/(x²−3) is an algebraic fraction, whereas √x/(x+1) is not, because the numerator is not a polynomial. The fraction is called proper if the degree of the numerator is less than the degree of the denominator; otherwise it is improper.

代数分式是分子和分母均为多项式的分式。例如 (2x+1)/(x²−3) 是代数分式，而 √x/(x+1) 则不是，因为分子不是多项式。如果分子的次数小于分母的次数，则称为真分式；否则为假分式。

2. Simplifying Algebraic Fractions | 化简代数分式

To simplify an algebraic fraction, factorise both the numerator and the denominator fully, then cancel any common factors. Always state the values of the variable for which the original denominator would be zero, as these are excluded from the domain. For instance, simplify (x²−4)/(x²−x−2): factorise to (x−2)(x+2)/[(x−2)(x+1)]. Cancel (x−2) to get (x+2)/(x+1), with x≠2, x≠−1.

化简代数分式时，应将分子和分母分别彻底因式分解，然后约去公因式。同时必须注明使原分母为零的变量取值，这些值需要从定义域中排除。例如，化简 (x²−4)/(x²−x−2)：因式分解得 (x−2)(x+2)/[(x−2)(x+1)]，约去 (x−2) 得 (x+2)/(x+1)，并注明 x≠2 且 x≠−1。

3. Multiplying and Dividing Algebraic Fractions | 代数分式的乘法与除法

Multiplication of algebraic fractions follows the same principle as numerical fractions: multiply the numerators together and multiply the denominators together, then simplify by cancelling common factors. It is often easier to cancel any common factors before multiplying – cross-cancelling – to keep the expressions manageable. For example, (x+3)/(x²−1) × (x−1)/(2x+6) = (x+3)/((x−1)(x+1)) × (x−1)/(2(x+3)) = 1/[2(x+1)] after cancelling (x+3) and (x−1). Division is performed by multiplying by the reciprocal of the divisor.

代数分式的乘法遵循与数值分式相同的原则：分子相乘、分母相乘，然后通过约分进行化简。通常在相乘之前先约分（交叉约分）能使表达式更简洁。例如 (x+3)/(x²−1) × (x−1)/(2x+6) = (x+3)/[(x−1)(x+1)] × (x−1)/[2(x+3)]，约去 (x+3) 和 (x−1) 后得 1/[2(x+1)]。除法运算则通过乘以除数的倒数来完成。

4. Adding and Subtracting Algebraic Fractions | 代数分式的加法与减法

Adding or subtracting algebraic fractions requires a common denominator. First, factorise each denominator to identify the lowest common denominator (LCD). Rewrite each fraction as an equivalent fraction with the LCD, combine the numerators, then simplify the resulting numerator and cancel if possible. For 1/(x+2) + 3/(x−1), the LCD is (x+2)(x−1). Thus 1/(x+2) becomes (x−1)/[(x+2)(x−1)] and 3/(x−1) becomes 3(x+2)/[(x+2)(x−1)]. Adding gives (x−1+3x+6)/[(x+2)(x−1)] = (4x+5)/[(x+2)(x−1)].

代数分式的加减法运算需要先找到公分母。首先将每个分母因式分解，以确定最低公分母（LCD）。然后将每个分式化为以最低公分母为分母的等值分式，合并分子，最后化简所得分子，并尽可能约分。例如 1/(x+2) + 3/(x−1)，LCD 为 (x+2)(x−1)，于是 1/(x+2) 化为 (x−1)/[(x+2)(x−1)]，3/(x−1) 化为 3(x+2)/[(x+2)(x−1)]，相加得 (x−1+3x+6)/[(x+2)(x−1)] = (4x+5)/[(x+2)(x−1)]。

5. Improper Fractions and Mixed Expressions | 假分式与带分式

When the numerator’s degree is greater than or equal to the denominator’s degree, the fraction is improper. In such cases, we perform polynomial long division (or synthetic division) to express the improper fraction as a polynomial plus a proper fraction. This is essential before partial fraction decomposition. For example, divide (x³+2x²−1)/(x+1): long division yields x²+x−1 with remainder 0, so the fraction simplifies exactly to x²+x−1. If there is a non‑zero remainder, the result is Q(x) + R(x)/D(x).

当分子的次数大于或等于分母的次数时，该分式为假分式。此时需要进行多项式长除法（或综合除法），将假分式表示为一个多项式与一个真分式的和。这在部分分式分解之前是必不可少的一步。例如，计算 (x³+2x²−1)/(x+1)：长除法得到商式 x²+x−1，余数为 0，因此假分式可化简为多项式 x²+x−1。若存在非零余式，则结果写成 Q(x) + R(x)/D(x) 的形式。

6. Partial Fractions – Distinct Linear Factors | 部分分式 – 不同线性因子

Partial fraction decomposition reverses the addition process, expressing a single rational function as a sum of simpler fractions. For a proper fraction whose denominator is a product of distinct linear factors, we write it as a sum of terms of the form A/(linear factor). For instance, decompose 5/((x−1)(x+2)): set 5/((x−1)(x+2)) ≡ A/(x−1) + B/(x+2). Multiply through by the denominator to get 5 ≡ A(x+2) + B(x−1). Substituting suitable x values (x=1, x=−2) gives A=5/3, B=−5/3, so the decomposition is (5/3)/(x−1) − (5/3)/(x+2).

部分分式分解是加法过程的逆运算，它将一个有理函数表示为若干更简单分式的和。对于分母为不同线性因子乘积的真分式，我们将其写成 A/(线性因子) 形式的各项之和。例如分解 5/((x−1)(x+2))：设 5/((x−1)(x+2)) ≡ A/(x−1) + B/(x+2)，两边同乘分母得 5 ≡ A(x+2) + B(x−1)。代入适当的 x 值（x=1, x=−2）解得 A=5/3, B=−5/3，因此分解结果为 (5/3)/(x−1) − (5/3)/(x+2)。

7. Partial Fractions – Repeated Linear Factors | 部分分式 – 重复线性因子

If the denominator contains a repeated linear factor, say (x+1)², the partial fraction expansion includes terms with all powers of that factor up to the multiplicity. For a proper fraction with (x+1)² in the denominator, the form is A/(x+1) + B/(x+1)². For example, to decompose (2x+3)/(x+1)², write (2x+3)/(x+1)² ≡ A/(x+1) + B/(x+1)². Multiply by (x+1)² to obtain 2x+3 ≡ A(x+1) + B. Equating coefficients or substituting x=−1 gives B=1, then compare coefficients of x to get A=2. Thus (2x+3)/(x+1)² = 2/(x+1) + 1/(x+1)².

如果分母中含有重复线性因子，例如 (x+1)²，则部分分式展开式中需要包含该因子的所有幂次项。对于分母含 (x+1)² 的真分式，展开形式为 A/(x+1) + B/(x+1)²。例如分解 (2x+3)/(x+1)²：设 (2x+3)/(x+1)² ≡ A/(x+1) + B/(x+1)²，两边同乘 (x+1)² 得 2x+3 ≡ A(x+1) + B。代入 x=−1 得 B=1，再比较 x 的系数可得 A=2。因此 (2x+3)/(x+1)² = 2/(x+1) + 1/(x+1)²。

8. Partial Fractions – Irreducible Quadratic Factors | 部分分式 – 不可约二次因子

When the denominator includes an irreducible quadratic factor (one that cannot be factorised into real linear factors), the corresponding partial fraction term has a linear numerator: (Ax + B)/(quadratic). For example, decompose (2x)/((x²+1)(x−2)). The form is (Ax+B)/(x²+1) + C/(x−2). Multiply through, equate coefficients or substitute values to find A, B and C. The solution yields A=0.4, B=0.8, C=0.8, so (2x)/((x²+1)(x−2)) = (0.4x+0.8)/(x²+1) + 0.8/(x−2).

当分母包含不可约二次因子（即无法分解为实线性因子的二次式）时，对应的部分分式项的分子应为线性式：(Ax + B)/(二次式)。例如分解 (2x)/((x²+1)(x−2))：其形式为 (Ax+B)/(x²+1) + C/(x−2)。两边乘以分母，通过比较系数或代入特定值求出 A、B、C。解得 A=0.4, B=0.8, C=0.8，从而 (2x)/((x²+1)(x−2)) = (0.4x+0.8)/(x²+1) + 0.8/(x−2)。

9. Solving Equations Involving Algebraic Fractions | 解含有代数分式的方程

To solve an equation that contains algebraic fractions, start by finding the lowest common denominator of all the fractions. Multiply every term by this LCD to clear the denominators, being careful to multiply the whole of each side. Solve the resulting polynomial equation. After finding the solutions, you must check that none of them makes any original denominator zero; such extraneous solutions must be discarded. For instance, solve x/(x−1) + 1/(x+1) = 2. The LCD is (x−1)(x+1). Multiplying gives x(x+1) + 1(x−1) = 2(x²−1). Simplify to x²+2x−3=0, giving x=1 or x=−3. However, x=1 is invalid because it makes the first denominator zero. The only solution is x=−3.

在解含有代数分式的方程时，首先找出所有分式的最低公分母。用这个最低公分母乘以方程两边的每一项以去分母，注意是整体相乘。解出得到的多项式方程。求出解后，必须验证这些解是否会使原方程中任何一个分母为零；这样的增根应当舍去。例如解方程 x/(x−1) + 1/(x+1) = 2：最低公分母为 (x−1)(x+1)，两边相乘得 x(x+1) + 1(x−1) = 2(x²−1)，化简得 x²+2x−3=0，解得 x=1 或 x=−3。但 x=1 会使第一个分母为零，故舍去，唯一解为 x=−3。

10. Applications to Functions and Graphs | 函数与图像中的应用

Algebraic fractions appear in rational functions. Simplifying a rational function or rewriting an improper fraction as a polynomial plus a proper fraction reveals asymptotic behaviour. For example, writing f(x) = (x²+2x)/(x−1) as x+3 + 3/(x−1) shows that as x→±∞, f(x) behaves like y=x+3, giving an oblique asymptote. The term 3/(x−1) indicates a vertical asymptote at x=1. Such skills are essential for curve sketching in topics like P3 or Further Maths.

代数分式出现在有理函数中。化简有理函数或将假分式改写为多项式加真分式的形式可以揭示渐近行为。例如将 f(x) = (x²+2x)/(x−1) 写成 x+3 + 3/(x−1) 后可以看出，当 x→±∞ 时，f(x) 的表现类似于 y=x+3，即存在斜渐近线；而项 3/(x−1) 说明存在垂直渐近线 x=1。这些技巧对于 P3 或进阶数学中的曲线草图绘制至关重要。

11. Common Mistakes and Pitfalls | 常见错误与陷阱

Students often forget to state domain restrictions after simplifying, which may lose marks in Edexcel exams. Another common error is incorrectly cancelling terms when they are not factors; for instance, in (x+2)/(x²+2), you cannot cancel x+2 because the denominator is not factorised. When solving equations, failing to check for extraneous solutions can lead to incorrect final answers. In partial fractions, always ensure the fraction is proper before decomposing, and use the correct form for repeated or quadratic factors.

学生们常常在化简之后忘记注明定义域限制，这在 Edexcel 考试中可能会失分。另一个常见错误是在非因式的情况下误约分，例如在 (x+2)/(x²+2) 中就不能约去 x+2，因为分母没有因式分解。解方程时若不检查增根，可能导致最后答案错误。在部分分式分解中，务必在分解前确保分式为真分式，并对重复因子或二次因子使用正确的形式。

12. Summary and Exam Tips | 小结与考试提示

Mastering algebraic fractions is a gateway skill for higher-level algebra in A‑Level Mathematics. Always start by factoring; use long division for improper fractions; be systematic when finding a common denominator; and verify your solutions. In Edexcel exams, algebraic fractions appear in Pure Mathematics units (e.g. P2, P3, P4) and in questions on integration, binomial expansion and proofs. Careful algebraic manipulation, clear presentation and domain checks will consistently earn you full marks.

熟练掌握代数分式是进入 A‑Level 数学更高层次代数学习的必经之路。务必先进行因式分解；对假分式使用长除法；找公分母时要有条理；并验证所得解。在 Edexcel 考试中，代数分式出现在纯数单元（如 P2、P3、P4）以及积分、二项式展开和证明题中。谨慎的代数操作、清晰的表达和定义域校验，将帮助你稳拿满分。

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Algebraic Fractions | 代数分式