📚 Rationalism in A-Level Maths: Rational Expressions and Rationalising Denominators | A-Level 数学中的理性思维：有理式与分母有理化
Rationalism in A-Level Mathematics is not a separate topic in the Edexcel specification; rather, it describes the logical, rule-based reasoning that underpins all algebraic work. In this revision guide we focus on rational expressions and rationalising denominators, two areas where rational thinking and formal manipulation combine.
在 A-Level 数学中，“理性主义”并非 Edexcel 考纲中的独立主题，而是指支撑所有代数运算的逻辑推理与规则意识。本复习指南聚焦有理式与分母有理化，这两个部分最能体现理性思维与形式化操作的结合。
1. Rationalism as Rule-Based Reasoning | 理性主义是规则驱动的推理
In Edexcel A-Level Mathematics, rationalism is best understood as the discipline of following formal algebraic rules and justifying every transformation. You do not guess results; you derive them from definitions such as equality, factoring, common denominators and polynomial arithmetic.
在 Edexcel A-Level 数学中，理性主义最好理解为遵循形式化代数规则并为每一步变换提供依据的学科习惯。你不能猜测结果，而是从等式、因式分解、公分母和多项式运算等定义出发推导结果。
2. What Is a Rational Expression? | 什么是有理式
A rational expression is a fraction whose numerator and denominator are polynomials. It can be written as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Rational expressions appear in algebraic fractions, partial fractions and many graph-sketching questions.
有理式是分子和分母均为多项式的分式。它可以写成 f(x) = P(x)/Q(x)，其中 Q(x) ≠ 0。有理式出现在代数分式、部分分式以及许多作图题中。
f(x) = P(x)/Q(x), Q(x) ≠ 0
| Expression | Reason |
| (2x + 1)/(x − 4) | Numerator and denominator are polynomials |
| 3/(x² + 1) | Polynomial ratio with constant numerator |
| √(x + 1)/(x − 3) | Not rational: numerator contains a square root |
3. Domain and Excluded Values | 定义域与排除值
The domain of a rational expression is all real x for which the denominator is not zero. Set Q(x) = 0 and exclude those values before simplifying; otherwise you may lose restrictions.
有理式的定义域是使分母不为零的所有实数 x。应先解 Q(x) = 0 并排除这些值，再进行化简；否则可能丢失限制条件。
For f(x) = (x + 2)/(x − 5), set x − 5 = 0 ⇒ x = 5, so domain is x ∈ ℝ, x ≠ 5
4. Simplifying Rational Expressions | 化简有理式
To simplify, factor both numerator and denominator completely and cancel common polynomial factors. You must record excluded values from the original expression even if they disappear after cancellation.
化简时需将分子和分母完全因式分解，然后约去公因式。即使某些取值在约分后消失，也必须记录原表达式中的排除值。
(x² − 9)/(x − 3) = ((x − 3)(x + 3))/(x − 3) = x + 3, x ≠ 3
5. Multiplying and Dividing Rational Expressions | 有理式的乘法与除法
For multiplication, multiply numerators and denominators directly after factorisation. For division, invert the second fraction and multiply. Always cancel common factors before expanding, and state restrictions for any denominator that could be zero.
乘法在因式分解后直接将分子相乘、分母相乘；除法先取第二个分式的倒数再相乘。应始终在展开前约去公因式，并声明所有可能为零的分母的限制。
A/B × C/D = AC/BD
A/B ÷ C/D = A/B × D/C
6. Adding and Subtracting Rational Expressions | 有理式的加法与减法
To add or subtract rational expressions, find the lowest common denominator, rewrite each fraction, combine numerators, and simplify. Factor the denominators first to identify the LCD efficiently.
加减有理式时，先求最低公分母，将每个分式通分后合并分子并化简。先对分母因式分解，可以更有效地确定最低公分母。
1/(x − 1) + 2/(x + 1) = (x + 1 + 2x − 2)/((x − 1)(x + 1)) = (3x − 1)/((x − 1)(x + 1))
7. Rationalising Denominators with Surds | 含根式分母的有理化
A denominator containing a surd such as √2 is considered unsimplified. Multiply numerator and denominator by the same surd so the denominator becomes rational.
分母含有如 √2 这样的根式时，通常视为未化简。将分子和分母同乘以该根式，使分母变为有理数。
1/√3 = (1 × √3)/(√3 × √3) = √3/3
For a fraction like 5/(2√5), multiply by √5/√5 to obtain 5√5/(2 × 5) = √5/2.
对于 5/(2√5) 这类分式，乘以 √5/√5 得到 5√5/(2 × 5) = √5/2。
8. Rationalising Using Conjugates | 使用共轭式进行有理化
For a binomial denominator such as a + b√c, multiply by its conjugate a − b√c. This uses the difference of two squares to remove the square root from the denominator.
对于 a + b√c 这类二项式分母，应乘以其共轭式 a − b√c。这利用平方差公式消去分母中的根号。
(√3 + 1)/(√3 − 1) = ((√3 + 1)²)/((√3)² − 1²) = (4 + 2√3)/2
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