📚 Multiplying and Dividing Algebraic Fractions | 乘除代数分式
Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions such as x, x², or x+3. Multiplying and dividing these fractions is a fundamental topic in IGCSE Mathematics, particularly for Edexcel. It combines your skills in factorising polynomials, identifying common factors, and simplifying rational expressions. A strong grasp of these operations will help you solve equations, simplify complex expressions, and perform better in algebra-related problems.
代数分式是指分子、分母或两者含有代数表达式（如 x、x² 或 x+3）的分式。乘除这些分式是IGCSE数学（尤其是Edexcel考试）中的基础主题。它结合了多项式因式分解、识别公因式和化简有理表达式的技能。牢固掌握这些运算将有助于解方程、化简复杂表达式，并在与代数相关的问题中取得更好成绩。
1. Recap: What Are Algebraic Fractions? | 回顾：什么是代数分式？
Just as with numerical fractions, you can multiply and divide algebraic fractions by following similar rules. The key difference is that you often need to factorise the numerators and denominators first to identify common factors that can be cancelled. For example, in a fraction like (x² - 4)/(x+2), you can factor the numerator as (x+2)(x-2) and then cancel the common factor (x+2), leaving x-2.
与数值分数类似，你可以通过类似的规则乘除代数分式。关键区别在于，你通常需要先对分子和分母进行因式分解，以识别可以约去的公因式。例如，对于像 (x² - 4)/(x+2) 这样的分式，你可以将分子因式分解为 (x+2)(x-2)，然后约去公因式 (x+2)，得到 x-2。
The same principle applies when multiplying or dividing: factorising first makes the process much easier and helps avoid errors. An algebraic fraction is undefined for any value of the variable that makes any denominator zero, a point we will revisit later.
在进行乘法或除法时，同样的原则也适用：先因式分解会使过程容易得多，并有助于避免错误。代数分式对于任何使得分母为零的变量值都是无定义的，我们稍后会再次强调这一点。
2. The First Rule: Factorise Everything! | 首要规则：因式分解一切！
Before you multiply or divide any algebraic fractions, always factorise every numerator and every denominator completely. This includes taking out common factors, using the difference of two squares, and factoring quadratic trinomials. For instance, factorise 2x² + 6x into 2x(x+3).
在乘除任何代数分式之前，务必将每个分子和每个分母完全因式分解。这包括提取公因式、利用平方差公式以及分解二次三项式。例如，将 2x² + 6x 分解为 2x(x+3)。
Why is this step so crucial? Because only after factorising can you see the common factors that can be cancelled. If you multiply first and then try to simplify, you may end up with a much larger expression that is harder to factor. Factorising also reveals any hidden restrictions on the variable.
为什么这一步如此重要？因为只有在因式分解之后，你才能看到可以约去的公因式。如果你先相乘再尝试化简，可能会得到一个更大的表达式，从而更难分解。因式分解还能揭示变量隐藏的限制条件。
Always check for the highest common factor (HCF) first, then look for patterns like x² - a² or ax² + bx + c. This habit will earn you marks for showing working and prevent basic mistakes.
总是先检查最高公因式 (HCF)，然后寻找诸如 x² - a² 或 ax² + bx + c 的模式。养成这个习惯不仅可以展示解题步骤得分，还能防止低级错误。
3. Multiplying Algebraic Fractions | 乘代数分式
To multiply two algebraic fractions, follow these steps: 1. Factorise all numerators and denominators. 2. Cancel any common factor that appears in any numerator and any denominator. 3. Multiply the remaining factors in the numerators together to form the new numerator. 4. Multiply the remaining factors in the denominators together to form the new denominator. 5. Check if the result can be simplified further.
要乘两个代数分式，请遵循以下步骤：1. 分解所有分子和分母。2. 约去出现在任何分子和任何分母中的任何公因式。3. 将分子中剩余的因式相乘，形成新的分子。4. 将分母中剩余的因式相乘，形成新的分母。5. 检查结果是否还能进一步化简。
and (x-1). The numerator becomes 1, the denominator becomes (x+1) × 3. Result: 1/[3(x+1)].
例如：化简 (x+2)/(x² - 1) × (x - 1)/(3x+6)。首先因式分解：(x+2)/[(x-1)(x+1)] × (x-1)/[3(x+2)]。约去公因式 (x+2) 和 (x-1)。分子变为 1，分母变为 (x+1) × 3。结果为 1/[3(x+1)]。
(x+2)/(x²-1) × (x-1)/(3x+6) = 1/[3(x+1)]
A common mistake is to cancel terms that are not factors. For example, in (x+2)/(x+3) you cannot cancel the 'x'
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