📚 Rational Expressions and Functions | 有理表达式与函数
A rational expression is an algebraic fraction in which both the numerator and the denominator are polynomials. The topic may look simple at first, but it threads through many Edexcel A-Level Mathematics skills: simplifying algebra, finding domains, solving equations, and integrating algebraic fractions.
有理表达式是一种分子和分母都为多项式的代数分式。这个主题表面上简单，但它贯穿 Edexcel A-Level 数学的许多技能：代数化简、求定义域、解方程以及对代数分式积分。
1. What Is a Rational Expression? | 什么是有理表达式？
A rational expression has the form P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) is not identically zero. For example, (x² + 3x + 2)/(x – 1), 5/(x + 2) and (2x³ – 4x + 7)/(x² + 1) are all rational expressions.
有理表达式的形式为 P(x)/Q(x)，其中 P(x) 和 Q(x) 都是多项式，且 Q(x) 不恒等于零。例如 (x² + 3x + 2)/(x – 1)、5/(x + 2) 和 (2x³ – 4x + 7)/(x² + 1) 都是有理表达式。
An expression is not rational if either the numerator or the denominator is not a polynomial. For instance, √x/(x + 1), sin x/(x² + 1) and |x|/(x – 3) are not rational expressions because they contain non-polynomial parts.
如果分子或分母不是多项式，则该表达式不是有理表达式。例如 √x/(x + 1)、sin x/(x² + 1) 和 |x|/(x – 3) 都不是有理表达式，因为它们包含非多项式部分。
In the Edexcel A-Level course, rational expressions appear in partial fractions, equation solving, curve sketching and integration. Being confident with factorising polynomials is therefore essential before studying this topic in depth.
在 Edexcel A-Level 课程中，有理表达式出现在部分分式、方程求解、曲线绘制和积分中。因此，在深入学习本主题之前，熟练地对多项式进行因式分解是至关重要的。
2. Domain and Excluded Values | 定义域与排除值
Because division by zero is undefined, a rational expression is only valid when its denominator is not zero. To find the domain, set the denominator equal to zero and exclude those values of x.
由于除以零没有定义，有理表达式只有在分母不为零时才有意义。求定义域时，令分母等于零，并排除这些 x 值。
For example, if f(x) = (x + 1)/(x – 2), then the denominator is zero when x – 2 = 0, so x = 2 is excluded. The domain is x ∈ ℝ, x ≠ 2.
例如，如果 f(x) = (x + 1)/(x – 2)，那么当 x – 2 = 0 时分母为零，因此 x = 2 被排除。定义域为 x ∈ ℝ，x ≠ 2。
An important exam point is that even when a common factor is cancelled, the original domain restriction does not disappear. If g(x) = (x² – 1)/(x² – x – 2), factorising gives (x – 1)(x + 1)/((x – 2)(x + 1)). After cancelling (x + 1), we get (x – 1)/(x – 2), but the original expression was undefined at x = 2 and x = -1, so both must be excluded.
一个重要的考点是，即使约去了公因式，原来的定义域限制也不会消失。若 g(x) = (x² – 1)/(x² – x – 2)，因式分解后得到 (x – 1)(x + 1)/((x – 2)(x + 1))。约去 (x + 1) 后得到 (x – 1)/(x – 2)，但原表达式在 x = 2 和 x = -1 处没有定义，因此这两个值都必须排除。
3. Simplifying Rational Expressions | 化简有理表达式
To simplify a rational expression, factorise both the numerator and the denominator fully, then cancel any common polynomial factor. Cancellation is only valid for multiplication, never for addition or subtraction.
要化简有理表达式，需要将分子和分母完全因式分解，然后约去公共的多项式因式。约分只对乘法有效，对加法或减法无效。
For instance, simplify (x² + x – 2)/(x² – 1). Factorising gives (x + 2)(x – 1)/((x – 1)(x + 1)). The common factor (x – 1) can be cancelled, giving:
例如，化简 (x² + x – 2)/(x² – 1)。因式分解得到 (x + 2)(x – 1)/((x – 1)(x + 1))。公共因式 (x – 1) 可以约去，得到：
(x² + x – 2)/(x² – 1) = (x + 2)/(x + 1), x ≠ 1, x ≠ -1
The final answer must state the excluded values. A common error is to write (x + 2)/(x + 1) as 2/1 or 3/2 by cancelling individual terms; this is incorrect because x is not a factor of the entire numerator or denominator.
最终答案必须写出被排除的值。一个常见错误是把 (x + 2)/(x + 1) 写成 2/1 或 3/2，也就是约去单独的项；这是不正确的，因为 x 不是整个分子或分母的因式。
4. Multiplying and Dividing Rational Expressions | 有理表达式的乘除
When multiplying rational expressions, factorise all numerators and denominators first, cancel common factors across the fractions, then multiply the remaining factors. This is usually easier than multiplying first and then factorising a much larger polynomial.
有理表达式相乘时，先对所有分子和分母进行因式分解，跨分式约去公共因式，然后再乘以剩余因式。这通常比先相乘再对更大的多项式进行因式分解更容易。
For example, simplify (x² – 4)/(x + 2) × (x + 1)/(x – 2). Factorising x² – 4 gives (x – 2)(x + 2), so the product becomes:
例如，化简 (x² – 4)/(x + 2) × (x + 1)/(x – 2)。因式分解 x² – 4 得到 (x – 2)(x + 2)，因此乘积变为：
(x – 2)(x + 2)/(x + 2) × (x + 1)/(x – 2) = x + 1
The restrictions are x ≠ -2 and x ≠ 2 from the original denominators, even though the simplified result is just x + 1.
根据原始分母，限制条件为 x ≠ -2 和 x ≠ 2，尽管化简结果只是 x + 1。
For division, rewrite the problem as multiplication by the reciprocal. For example, (x² – 1)/(x + 3) ÷ (x – 1)/(x + 2) becomes (x – 1)(x + 1)/(x + 3) × (x + 2)/(x – 1), which simplifies to (x + 1)(x + 2)/(x + 3), with x ≠ 1, -3, -2.
除法时，将问题改写为乘以倒数。例如 (x² – 1)/(x + 3) ÷ (x – 1)/(x + 2) 变为 (x – 1)(x + 1)/(x + 3) × (x + 2)/(x – 1)，化简为 (x + 1)(x + 2)/(x + 3)，其中 x ≠ 1、-3、-2。
5. Adding and Subtracting Rational Expressions | 有理表达式的加减
To add or subtract rational expressions, first find a common denominator. If the denominators are distinct linear factors, the common denominator is usually their product. Then combine the numerators and simplify.
有理表达式相加或相减时，首先找到公共分母。如果分母是不同的线性因式，公共分母通常是它们的乘积。然后合并分子并化简。
For example, simplify 2/(x – 1) + 3/(x + 2). The common denominator is (x – 1)(x + 2). Writing both fractions over this denominator gives:
例如，化简 2/(x – 1) + 3/(x + 2)。公共分母是 (x – 1)(x + 2)。将两个分式都写成以该公分母为分母：
2(x + 2)/((x – 1)(x + 2)) + 3(x – 1)/((x – 1)(x + 2)) = (5x + 1)/((x – 1)(x + 2))
This result is useful in the reverse direction when splitting a rational expression into partial fractions. Always state that the original expression is undefined when x = 1 or x = -2.
这个结果在将有理表达式拆分为部分分式时非常有用。始终要说明原表达式在 x = 1 或 x = -2 处没有定义。
For subtraction, the same principle applies, but be careful to distribute the negative sign across the entire numerator of the fraction being subtracted. Brackets help avoid sign errors.
对于减法，原理相同，但要小心将负号分配到被减分式的整个分子上。使用括号有助于避免符号错误。
6. Partial Fractions | 部分分式
Partial fractions are used to express a single rational expression as a sum of simpler fractions. This technique is especially important for integration and sometimes for series expansion.
部分分式用于将一个有理表达式表示为若干个更简单分式的和。这种技巧对积分以及有时对级数展开特别重要。
A rational expression is proper if the degree of the numerator is less than the degree of the denominator. If it is improper, divide first so that the expression becomes a polynomial plus a proper remainder.
如果分子的次数小于分母的次数，则有理表达式是真分式。如果是假分式，应先进行多项式除法，使表达式变为多项式加真余式。
For distinct linear factors, write:
对于不同的线性因式，写成：
(5x + 1)/((x – 1)(x + 2)) = A/(x – 1) + B/(x + 2)
Multiplying through by the denominator gives 5x +
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