📚 Rational Expressions and Functions | 有理表达式与有理函数
A rational expression is a fraction in which the numerator and the denominator are polynomials. Understanding how to simplify, manipulate and graph these expressions is a core skill in A‑Level mathematics. This article covers the key techniques you will need for the Edexcel Mathematics specification, from basic simplification to partial fractions, asymptotes and integration.
有理表达式是分子和分母均为多项式的分式。掌握如何化简、操作以及绘制这类表达式是 A‑Level 数学中的核心技能。本文涵盖 Edexcel 数学考纲所需的关键技巧，从基本的化简到部分分式、渐近线和积分。
1. What is a Rational Expression? | 什么是有理表达式？
A rational expression is any expression of the form P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. Examples include (x²+3x+2)/(x+1) and 5/(x-2). The domain excludes any value of x that makes the denominator zero.
有理表达式是任何形如 P(x)/Q(x) 的表达式，其中 P(x) 和 Q(x) 是多项式且 Q(x) ≠ 0。例如 (x²+3x+2)/(x+1) 和 5/(x-2)。其定义域必须排除所有使分母为零的 x 值。
When the degree of the numerator is less than the degree of the denominator, the expression is said to be proper; otherwise it is improper. Improper rational expressions can be rewritten as a polynomial plus a proper rational expression, often using algebraic division.
当分子的次数低于分母的次数时，该表达式称为真分式；否则为假分式。假有理表达式可以改写为一个多项式加上一个真有理表达式，通常使用代数除法。
2. Simplifying Rational Expressions | 化简有理表达式
factors. For example, (x²-4)/(x²-x-6) = (x-2)(x+2)/[(x-3)(x+2)] = (x-2)/(x-3), for x ≠ -2.
要化简有理表达式，需将分子和分母完全因式分解，然后约去所有公因式。例如 (x²-4)/(x²-x-6) = (x-2)(x+2)/[(x-3)(x+2)] = (x-2)/(x-3)，其中 x ≠ -2。
Always state the restrictions on x. Cancelling a factor like (x+2) is valid only if that factor is not zero; otherwise you are dividing by zero. The simplified expression is only equivalent to the original when those excluded values are noted.
务必说明 x 的限制条件。约去 (x+2) 这样的因式只有在它不为零时才是有效的，否则相当于除以零。只有标注了这些被排除的值，化简后的表达式才与原式等价。
3. Multiplying and Dividing Rational Expressions | 有理表达式的乘法与除法
(x-3)(x+5)]. Always factorise first and cancel before multiplying to keep the work simple.
两个有理表达式相乘，只需将分子相乘作为新的分子，分母相乘作为新的分母。例如 (x+1)/(x-3) × (2x)/(x+5) = (2x(x+1))/[(x-3)(x+5)]。务必先因式分解并在乘法前约分，以保持计算简洁。
e, (x)/(x+2) ÷ (x²-4)/(3) = (x)/(x+2) × 3/[(x-2)(x+2)] = 3x/[(x+2)²(x-2)], with x ≠ -2, 2.
除法时，将第一个分式乘以第二个分式的倒数。即 (a/b) ÷ (c/d) = (a/b) × (d/c)。例如 (x)/(x+2) ÷ (x²-4)/(3) = (x)/(x+2) × 3/[(x-2)(x+2)] = 3x/[(x+2)²(x-2)]，且 x ≠ -2, 2。
4. Adding and Subtracting Rational Expressions | 有理表达式的加法与减法
ator is (x-1)(x+2). So the sum becomes [3(x+2)+2(x-1)]/[(x-1)(x+2)] = (5x+4)/[(x-1)(x+2)].
有理表达式的加法或减法需要找到公分母。先将每个分母因式分解，再构造最小公倍式。对于 3/(x-1) + 2/(x+2)，公分母为 (x-1)(x+2)，和变为 [3(x+2)+2(x-1)]/[(x-1)(x+2)] = (5x+4)/[(x-1)(x+2)]。
Be careful with signs, especially when a numerator contains more than one term and the subtraction sign affects the whole numerator. Write brackets if necessary to avoid mistakes, e.g. (x+3)/(x-2) - (2x-1)/(x) requires careful handling.
注意符号，特别是当分子包含不止一项而减号影响到整个分子时。必要时使用括号以避免错误，例如 (x+3)/(x-2) - (2x-1)/(x) 需要谨慎处理。
5. Partial Fractions – Linear Denominators | 部分分式 – 线性分母
Many rational expressions can be decomposed into a sum of simpler fractions, which is essential for integration. If the denominator splits into distinct linear factors, write the expression as the sum of constants over each factor. For example, for (3x+5)/[(x-1)(x+2)], assume form A/(x-1) + B/(x+2).
许多有理表达式可以分解为简单分式的和，这对于积分至关重要。如果分母可分解为不同的线性因式，则将该表达式写成每个因式上的常数之和。例如，对于 (3x+5)/[(x-1)(x+2)]，假设形式为 A/(x-1) + B/(x+2)。
Multiply through by the common denominator to obtain 3x+5 ≡ ind A and B. Here, substituting x=1 gives 8 = 3A, so A=8/3; x=-2 gives -1 = -3B, so B=1/3.
两边同乘公分母得 3x+5 ≡ A(x+2)+B(x-1)。通过比较系数或代入方便的 x 值求 A 和 B。此处代入 x=1 得 8 = 3A，故 A=8/3；x=-2 得 -1 = -3B，故 B=1/3。
6. Partial Fractions – Repeated and Quadratic Factors | 部分分式 – 重复与二次因式
When a linear factor is repeated, say (x+1)², the decomposition must include both A/(x+1) and B/(x+1)². For (4x+3)/(x+1)²(x-2), write A/(x+1) + B/(x+1)² + C/(x-2).
当线性因式重复出现，例如 (x+1)² 时，分解必须同时包含 A/(x+1) 和 B/(x+1)²。对于 (4x+3)/(x+1)²(x-2)，写成 A/(x+1) + B/(x+1)² + C/(x-2)。
If the denominator includes an irreducible quadratic factor such as (x²+1), the corresponding numerator is a linear expression, not just a constant. For (5x²+2)/[(x-1)(x²+4)], the form is A/(x-1) + (Bx+C)/(x²+4).
如果分母包含不可约二次因式，如 (x²+1)，对应的分子应是一次表达式，而非仅仅常数。对于 (5x²+2)/[(x-1)(x²+4)]，形式为 A/(x-1) + (Bx+C)/(x²+4)。
7. Rational Functions and Asymptotes | 有理函数与渐近线
(x+3), vertical asymptotes are x=2 and x=-3, assuming the numerator does not vanish there.
有理函数由有理表达式定义。垂直渐近线出现在分母为零且该因式不与分子约去的位置。对于 f(x) = (x+1)/(x-2)(x+3)，垂直渐近线为 x=2 和 x=-3，前提是分子在这些值上不为零。
nt of numerator)/(leading coefficient of denominator). If numerator degree is greater, there is no horizontal asymptote; instead you may have an oblique asymptote found by division.
水平渐近线取决于分子和分母的次数。若分子次数 < 分母次数，水平渐近线为 y=0。若次数相等，渐近线为 y = (分子首项系数)/(分母首项系数)。若分子次数更高，则无水平渐近线，但可通过除法获得斜渐近线。
8. Graphs of Rational Functions | 有理函数的图像
asymptote is x=2, horizontal asymptote y=3. Mark intercepts: set y=0 gives 1/(x-2) + 3 = 0 ⇒ 1/(x-2) = -3 ⇒ x-2 = -1/3 ⇒ x = 5/3; x-intercept is (5/3,0).
要绘制 y = 1/(x-2) + 3 的图像，可将其视为反比例函数向右移 2、向上移 3。垂直渐近线为 x=2，水平渐近线为 y=3。标注截距：令 y=0 得 1/(x-2) + 3 = 0 ⇒ 1/(x-2) = -3 ⇒ x-2 = -1/3 ⇒ x = 5/3，x 截距为 (5/3,0)。
For more complicated functions, find asymptotes, intercepts and test sign behaviour around vertical asymptotes. A sign table using intervals between asymptotes and intercepts helps to determine where the curve lies above or below the x-axis. The curve approaches but never crosses a vertical asymptote.
对于更复杂的函数，需找出渐近线、截距，并检测垂直渐近线附近的符号变化。利用渐近线与截距划分区间构造符号表，可以判断曲线在 x 轴上方还是下方。曲线无限靠近但永不穿过垂直渐近线。
9. Integration of Rational Functions | 有理函数的积分
or example, ∫ (3x+5)/[(x-1)(x+2)] dx can be integrated after decomposition to give (8/3)ln|x-1| + (1/3)ln|x+2| + C.
积分有理函数通常需要先将其转化为可直接积分的形式。当分母可因式分解时，使用部分分式。例如 ∫ (3x+5)/[(x-1)(x+2)] dx 分解后可积分为 (8/3)ln|x-1| + (1/3)ln|x+2| + C。
and the proper part can be handled by partial fractions. Also, integrals of the form ∫ (f '(x))/f(x) dx = ln|f(x)| + C appear frequently.
对于假有理表达式，应首先进行多项式长除法，得到一个多项式加上真分式。多项式部分容易积分，真分式部分可用部分分式处理。此外，形如 ∫ (f '(x))/f(x) dx = ln|f(x)| + C 的形式经常出现。
10. Solving Equations with Rational Expressions | 解含有理表达式的方程
, leading to x = -7. Check that this solution does not make any original denominator zero.
要解诸如 2/(x+1) = 3/(x-2) 的方程，可找到公分母并两边相乘，或者当方程两边各只有一个分式时直接交叉相乘。此处交叉相乘得 2(x-2) = 3(x+1)，解得 x = -7。须检查该解是否使原分母为零。
If the equation has three or more terms, bring all terms to one side, combine into a single rational expression, and set the numerator equal to zero. Always reject extraneous roots that make the denominator zero.
如果方程含有三项或更多项，将所有项移到一边并合并为单一有理表达式，令分子为零求解。必须舍去使分母为零的增根。
11. Modelling with Rational Functions | 有理函数的建模应用
d at speed v is t = d/v, a rational function with horizontal asymptote t=0 (infinite speed – impossible) and vertical asymptote v=0 (speed zero – time becomes infinite).
有理函数可模拟许多涉及比率、反比关系和最优化的现实情境。例如，以速度 v 行驶固定距离 d 所需的时间为 t = d/v，这是一个有理函数，水平渐近线为 t=0（速度无限大——不可能），垂直渐近线为 v=0（速度为零——时间无穷大）。
ving yields a rational equation that eventually gives a quadratic. Such models appear frequently in exam questions; the key is to set up the rational relationship correctly.
当组合效率时，比如两个水管注满水箱，你可能会得到方程 1/x + 1/(x+3) = 1/2。求解产生一个有理方程，最终化为二次方程。此类模型常出现在考题中，关键在于正确建立有理关系式。
12. Common Mistakes and Exam Tips | 常见错误与应试技巧
One frequent error is cancelling terms instead of factors. You can cancel (x+2)/(x+2) but never cancel an x inside a sum. Also, forgetting to state the domain restrictions when cancelling can lead to incomplete answers. Always write ‘x ≠ …’ alongside your simplified expression.
一个常见错误是约掉项而非因式。可以约去 (x+2)/(x+2)，但绝不能约去加法中的 x。此外，约分时忘记说明定义域限制会导致答案不完整。务必在化简后写出 ‘x ≠ …’ 之类的说明。
In partial fractions, double-check your cover-up values and consider whether the fraction is proper. If not, perform division first. When sketching, do not forget to label asymptotes as dashed lines and mark any intercepts. Finally, integration of rational functions often relies on recognising the ln form – practice identifying f '(x)/f(x) quickly.
在部分分式中，仔细检查遮盖法所得的值，并判断是否为真分式，若不是则先做除法。绘图时不要忘记用虚线标注渐近线并标出截距。最后，有理函数的积分常依赖于识别对数形式——多加练习，快速识别 f '(x)/f(x)。
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