📚 Simplifying and Operations with Algebraic Fractions | 代数分式的化简与运算
An algebraic fraction is a fraction whose numerator and/or denominator contains an algebraic expression. In this guide, we focus on rational expressions of the form P/Q, where P and Q are polynomials and Q is not the zero polynomial.
代数分式是指分子或分母中含有代数表达式的分数。本指南重点讨论形如 P/Q 的有理表达式，其中 P 和 Q 都是多项式，且 Q 不为零多项式。
For example, (2x + 1)/(x − 3) and (x² − 4)/(x² + 5x + 6) are algebraic fractions. They follow the same rules as numerical fractions, but they also require factorisation and careful attention to the values for which the denominator is zero.
例如，(2x + 1)/(x − 3) 和 (x² − 4)/(x² + 5x + 6) 都是代数分式。它们遵循与数值分数相同的运算规则，但还需要进行因式分解，并特别注意使分母为零的那些取值。
1. What Is an Algebraic Fraction? | 什么是代数分式？
An algebraic fraction can be as simple as 3/x or as complicated as (x³ − 1)/(x² − 4x + 4). In every case, the denominator must not be zero for the expression to be defined.
代数分式可以像 3/x 一样简单，也可以像 (x³ − 1)/(x² − 4x + 4) 一样复杂。在任何情况下，分母都不能为零，否则表达式没有意义。
The same basic principle applies: if we multiply or divide both the numerator and the denominator by the same non-zero expression, the value of the fraction does not change. This principle allows us to simplify algebraic fractions.
这里遵循同样的基本原理：如果我们将分子和分母同时乘以或除以同一个非零表达式，分数的值不变。正是这一原理使我们能够化简代数分式。
2. The Golden Rule: Domain Restrictions | 黄金法则：定义域限制
Before simplifying any algebraic fraction, you must decide which values of the variable are allowed. The denominator of the original fraction must never be zero, so every value that makes any factor of the original denominator equal to zero must be excluded.
在化简任何代数分式之前，必须先确定变量的哪些取值是允许的。原分式的分母绝不能为零，因此所有使原分母中任一因式等于零的取值都必须排除。
For example, in the fraction (x − 2)/(x + 3), the denominator is zero when x = −3. Therefore we write the simplified result with the condition x ≠ −3.
例如，在分式 (x − 2)/(x + 3) 中，当 x = −3 时分母为零。因此，在写出化简结果时，必须附上条件 x ≠ −3。
For the fraction x/(x² − 1), factorise the denominator as x/((x − 1)(x + 1)). The excluded values are x = 1 and x = −1.
对于分式 x/(x² − 1)，将分母因式分解为 x/((x − 1)(x + 1))。排除的取值为 x = 1 和 x = −1。
x/(x² − 1) is defined only when x ≠ 1 and x ≠ −1.
x/(x² − 1) 仅在 x ≠ 1 且 x ≠ −1 时有定义。
3. Factorising Before Simplifying | 先因式分解再化简
To simplify an algebraic fraction, factorise the numerator and denominator completely, then cancel common factors. Cancelling can only be done between factors, never between individual terms.
化简代数分式时，应先将分子和分母完全因式分解，再约去公因式。约分只能发生在因式之间，绝不能作用于单独的项。
Consider the expression (x² − 1)/(x² + 2x + 1). Factorise the numerator as (x − 1)(x + 1) and the denominator as (x + 1)².
考虑表达式 (x² − 1)/(x² + 2x + 1)。分子因式分解为 (x − 1)(x + 1)，分母因式分解为 (x + 1)²。
(x² − 1)/(x² + 2x + 1) = ((x − 1)(x + 1))/((x + 1)(x + 1)) = (x − 1)/(x + 1), x ≠ −1.
(x² − 1)/(x² + 2x + 1) = ((x − 1)(x + 1))/((x + 1)(x + 1)) = (x − 1)/(x + 1)，x ≠ −1。
Notice that we still write x ≠ −1 even though the simplified form (x − 1)/(x + 1) is also undefined there. The original denominator requires this condition, and correct exam answers must include it.
注意，即使化简后的 (x − 1)/(x + 1) 在 x = −1 处也无定义，我们仍然要写上 x ≠ −1。原分母要求这一条件，考试中正确的答案必须包含它。
4. Multiplying Algebraic Fractions | 代数分式的乘法
When multiplying two algebraic fractions, multiply the numerators together and multiply the denominators together. It is often easier to factorise and cancel before multiplying, because the final expression becomes simpler.
两个代数分式相乘时，分子与分子相乘，分母与分母相乘。通常先因式分解并约分再相乘会更简单，因为最终表达式会更简洁。
For example, (2x/3) × (9/x²). Rewrite as (2x × 9)/(3 × x²). Since 9/3 = 3 and x/x² = 1/x, the result is 6/x, with x ≠ 0.
例如，(2x/3) × (9/x²)。改写为 (2x × 9)/(3 × x²)。因为 9/3 = 3，x/x² = 1/x，所以结果为 6/x，其中 x ≠ 0。
Another example: (x² − 9)/(x + 2) × (x + 2)/(x − 3). Factorise x² − 9 as (x − 3)(x + 3), then cancel (x + 2) and (x − 3).
另一个例子：(x² − 9)/(x + 2) × (x + 2)/(x − 3)。将 x² − 9 分解为 (x − 3)(x + 3)，然后约去 (x + 2) 和 (x − 3)。
(x² − 9)/(x + 2) × (x + 2)/(x − 3) = x + 3, x ≠ −2, 3.
(x² − 9)/(x + 2) × (x + 2)/(x − 3) = x + 3，x ≠ −2，3。
5. Dividing Algebraic Fractions | 代数分式的除法
To divide by an algebraic fraction, multiply by its reciprocal. This is the same rule used for numerical fractions, but you must also consider whether the divisor is zero.
除以一个代数分式，等于乘以它的倒数。这与数值分数的除法规则相同，但还需要判断除数是否为零。
For example, (x² − 4)/(x + 2) ÷ (x − 2)/(x + 3). First factorise x² − 4 as (x − 2)(x + 2), then multiply by the reciprocal of the divisor.
例如，(x² − 4)/(x + 2) ÷ (x − 2)/(x + 3)。先将 x² − 4 分解为 (x − 2)(x + 2)，再乘以除数的倒数。
((x − 2)(x + 2))/(x +
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