Multiplying and Dividing Algebraic Fractions | 代数分式的乘除

📚 Multiplying and Dividing Algebraic Fractions | 代数分式的乘除

Algebraic fractions are fractions that contain variables in their numerators or denominators. Multiplying and dividing them follows the same fundamental rules as numerical fractions, but with one crucial addition: you must factorise expressions fully and cancel common factors to reach the simplest form. This guide will walk you through every step, with worked examples tailored to the Edexcel IGCSE syllabus.

代数分式是分子或分母中含有变量的分数。它们的乘除遵循与数字分数相同的基本规则，但有一个关键补充：你必须完全因式分解表达式并约去公因子，才能得到最简形式。本指南将逐步引导你完成每一个步骤，并附有面向 Edexcel IGCSE 考纲的典型例题。

1. Basic Principles of Multiplying Algebraic Fractions | 代数分式乘法基本原则

To multiply two fractions, you multiply the numerators together and multiply the denominators together. For algebraic fractions, this means:

两个分数相乘时，将分子与分子相乘，分母与分母相乘。对于代数分式，这意味着：

(a/b) × (c/d) = ac / bd

For example, (2x/3) × (5/y) = (2x × 5) / (3 × y) = 10x / 3y. However, the product should always be simplified by cancelling any common factors between the numerator and denominator.

例如，(2x/3) × (5/y) = (2x × 5) / (3 × y) = 10x / 3y。然而，乘积应始终通过约去分子与分母之间的任何公因子来化简。

Always check whether any factor in the numerator of either fraction can cancel with a factor in the denominator of either fraction before or after multiplying. This prevents unnecessarily large expressions.

在相乘之前或之后，务必检查任一分子中的因子是否可与任一分母中的因子约分。这样可以避免产生不必要的大型表达式。

2. Simplifying Before Multiplying | 乘法前先约分

When multiplying algebraic fractions, it is much easier to cancel common factors before performing the multiplication. This reduces the size of the numbers and makes simplification at the end much quicker.

在乘法中，先约去公因子再进行乘法运算会容易得多。这能缩小数值，使最后的化简更快。

Consider (4x/9) × (3/2x). Instead of multiplying first, cancel: 4 and 2 share a factor of 2; 9 and 3 share a factor of 3; 4 and 2x share a factor of 2x? No — 4 and 2x share a factor of 2 only if we think of 2x as a product. Let’s do it carefully:

考虑 (4x/9) × (3/2x)。不要先相乘，而是约分：4 与 2 有公因子 2；9 与 3 有公因子 3；4 与 2x 有公因子 2。我们仔细来做：

Write (4x/9) × (3/2x) = (4x × 3) / (9 × 2x). Now the x in the numerator cancels with the x in the denominator: (4 × 3) / (9 × 2). Then 4/2 = 2, and 3/9 = 1/3, giving 2 × (1/3) = 2/3. Alternatively, divide 4 by 2 to get 2, and divide 3 by 9 to get 1/3, then 2 × 1 = 2 in the numerator and 1 × 3 = 3 in the denominator.

写出 (4x/9) × (3/2x) = (4x × 3) / (9 × 2x)。分子中的 x 与分母中的 x 约去：(4 × 3) / (9 × 2)。然后 4/2 = 2，3/9 = 1/3，得到 2 × (1/3) = 2/3。或者先将 4 除以 2 得 2，将 3 除以 9 得 1/3，那么分子为 2 × 1 = 2，分母为 1 × 3 = 3。

Always cancel vertically or diagonally (numerator of one fraction with denominator of the other), never horizontally (numerator with numerator or denominator with denominator).

总是垂直或对角约分（一个分数的分子与另一个分数的分母），绝不能水平约分（分子与分子、分母与分母）。

3. Multiplying Monomial Fractions | 单项式分式相乘

Monomial fractions have single-term numerators and denominators, such as (3x²/5) or (7y/2x). To multiply them, multiply the coefficients, multiply the variables, and apply the laws of indices.

单项式分式的分子和分母都是单项式，如 (3x²/5) 或 (7y/2x)。将它们相乘时，先乘系数，再乘变量，并应用指数法则。

Example: Simplify (2x³/5) × (15/4x²).

示例：化简 (2x³/5) × (15/4x²)。

Cancel first: 2 and 4 share a factor of 2, so 2 → 1 and 4 → 2. 15 and 5 share a factor of 5, so 15 → 3 and 5 → 1. Also x³ cancels with x² leaving x in the numerator: x³ / x² = x^(3−2) = x. Therefore:

先约分：2 与 4 有公因子 2，所以 2 → 1，4 → 2。15 与 5 有公因子 5，所以 15 → 3，5 → 1。同时 x³ 与 x² 约去，分子剩 x：x³ / x² = x^(3−2) = x。因此：

(2x³/5) × (15/4x²) = (1 × 3 × x) / (1 × 2) = 3x/2

Remember the index laws: xᵐ × xⁿ = x^(m+n) and xᵐ / xⁿ = x^(m−n). These are essential when simplifying products of algebraic fractions.

记住指数法则：xᵐ × xⁿ = x^(m+n)，xᵐ / xⁿ = x^(m−n)。在化简代数分式的乘积时，这些法则至关重要。

4. Multiplying Polynomial Fractions | 多项式分式相乘

When numerators or denominators contain polynomials, you must first factorise each polynomial completely. Then cancel common factors before multiplying.

当分子或分母含有多项式时，你必须先将每个多项式完全因式分解。然后在相乘之前约去公因子。

Example 1: Simplify (x² − 9)/(x + 3) × (2x + 6)/4.

示例 1：化简 (x² − 9)/(x + 3) × (2x + 6)/4。

Factorise: x² − 9 = (x − 3)(x + 3) using difference of squares. Also 2x + 6 = 2(x + 3). So:

因式分解：x² − 9 = (x − 3)(x + 3)（平方差公式）。同时 2x + 6 = 2(x + 3)。所以：

((x − 3)(x + 3))/(x + 3) × (2(x + 3))/4

Now cancel (x + 3) from the first fraction and the denominator of the first fraction. Also note the second fraction has (x + 3) in its numerator but there is no (x + 3) in the denominator of the second fraction, so it does not cancel yet. However, we can cancel the factor 2 with the 4: 2/4 = 1/2. Thus:

现在从第一个分式的分子和分母中约去 (x + 3)。注意第二个分式的分子中有 (x + 3)，但第二个分式的分母中没有 (x + 3)，所以它暂不能约。然而，我们可将因子 2 与 4 约分：2/4 = 1/2。因此：

(x − 3) × 1 × (x + 3)/2 = (x − 3)(x + 3)/2 = (x² − 9)/2

Actually, let’s recheck: after cancelling (x + 3) from the first fraction, we have (x − 3) × (2(x + 3))/4 = (x − 3)(x + 3)/2 = (x² − 9)/2. The final answer is (x² − 9)/2, but it is often cleaner to leave it as (x−3)(x+3)/2 or expand. On the IGCSE exam, either factored or expanded form is acceptable unless otherwise stated.

实际上，让我们重新检查：约去第一个分式的 (x + 3) 后，得到 (x − 3) × (2(x + 3))/4 = (x − 3)(x + 3)/2 = (x² − 9)/2。最终答案可以是 (x² − 9)/2，但通常保留为 (x−3)(x+3)/2 或展开均可。IGCSE 考试中，除非另有说明，因式形式或展开形式都算对。

Example 2: Simplify (x² + 5x + 6)/(x² − 4) × (x − 2)/(x + 3).

示例 2：化简 (x² + 5x + 6)/(x² − 4) × (x − 2)/(x + 3)。

Factorise: x² + 5x + 6 = (x + 2)(x + 3). The denominator x² − 4 = (x + 2)(x − 2). Therefore:

因式分解：x² + 5x + 6 = (x + 2)(x + 3)。分母 x² − 4 = (x + 2)(x − 2)。因此：

((x + 2)(x + 3))/((x + 2)(x − 2)) × (x − 2)/(x + 3) = 1

After cancelling (x + 2), (x − 2), and (x + 3) from numerator and denominator, everything cancels, leaving 1. This demonstrates why factorising first is so powerful.

约去分子和分母中的 (x + 2)、(x − 2) 和 (x + 3) 后，所有项都约去了，剩下 1。这说明了为什么先因式分解如此重要。

5. Dividing Algebraic Fractions | 代数分式除法

Dividing by a fraction is equivalent to multiplying by its reciprocal. In other words, keep the first fraction unchanged, change the division sign to multiplication, and flip the second fraction (numerator becomes denominator and denominator becomes numerator).

除以一个分数等于乘以它的倒数。换句话说，保持第一个分式不变，将除号改为乘号，并将第二个分式翻转（分子变分母，分母变分子）。

For algebraic fractions:

对于代数分式：

(a/b) ÷ (c/d) = (a/b) × (d/c) = ad / bc

Example: Simplify (3x/4) ÷ (x/8).

示例：化简 (3x/4) ÷ (x/8)。

Flip the second fraction: (3x/4) × (8/x). Now cancel x: (3 × 8)/4 = 24/4 = 6. So (3x/4) ÷ (x/8) = 6.

翻转第二个分式：(3x/4) × (8/x)。约去 x：(3 × 8)/4 = 24/4 = 6。所以 (3x/4) ÷ (x/8) = 6。

Be very careful: the reciprocal only applies to the fraction immediately after the division sign. If dividing by a whole number or a single term, write it as a fraction first.

要非常小心：倒数只适用于紧跟除号之后的分式。如果除以一个整数或单项式，先将它写成分数形式。

6. Dividing with Polynomials (Factorization) | 多项式因式分解后的除法

When dividing algebraic fractions that include polynomials, factorise every numerator and denominator completely, then flip the second fraction and cancel common factors.

当被除的代数分式含有多项式时，先将每个分子和分母完全因式分解，然后翻转第二个分式，再约去公因子。

Example: Simplify (x² − 1)/(x² − 2x + 1) ÷ (x + 1)/(x − 1).

示例：化简 (x² − 1)/(x² − 2x + 1) ÷ (x + 1)/(x − 1)。

Step 1: Factorise all parts.

第一步：因式分解所有部分。

x² − 1 = (x − 1)(x + 1). The denominator x² − 2x + 1 = (x − 1)². So the first fraction becomes ((x−1)(x+1))/((x−1)²). The second fraction is (x + 1)/(x − 1).

x² − 1 = (x − 1)(x + 1)。分母 x² − 2x + 1 = (x − 1)²。所以第一个分式变为 ((x−1)(x+1))/((x−1)²)。第二个分式是 (x + 1)/(x − 1)。

Step 2: Flip and multiply.

第二步：翻转并相乘。

((x−1)(x+1))/((x−1)²) × (x−1)/(x+1)

Step 3: Cancel. The factor (x − 1) cancels one power from the denominator and the numerator of the first fraction, and the (x − 1) in the second numerator cancels with the remaining (x − 1) in the denominator. The (x + 1) cancels too. Result:

第三步：约分。因子 (x − 1) 约去第一个分式分母中的一个幂和第二个分式分子中的 (x − 1)，分母中剩余的 (x − 1) 也可约去。(x + 1) 也约去。结果：

(x+1)/(x−1) × (x−1)/(x+1) = 1

Actually, let’s carefully recalculate. The first fraction simplifies to (x+1)/(x−1) after cancelling (x−1). Then ((x+1)/(x−1)) × ((x−1)/(x+1)) = 1. Yes, the result is 1.

实际上，让我们仔细重算。第一个分式约去 (x−1) 后化简为 (x+1)/(x−1)。然后 ((x+1)/(x−1)) × ((x−1)/(x+1)) = 1。是的，结果为 1。

7. Common Mistakes and Pitfalls | 常见错误与陷阱

Many students lose marks on algebraic fractions due to avoidable errors. Here are the most frequent pitfalls and how to avoid them.

许多学生在代数分式上失分，是因为一些可以避免的错误。以下是最常见的陷阱及避免方法。

Cancelling terms instead of factors: You can only cancel factors that are common to the entire numerator and denominator, not individual terms. For example, in (x + 2)/(x + 3), you cannot cancel x because x is a term, not a factor.
仅约项而非约因子：只能约去整个分子和分母共有的因子，不能约个别项。例如，在 (x + 2)/(x + 3) 中，不能约 x，因为 x 是项而非因子。
Forgetting to flip the second fraction when dividing: A very common error. Always rewrite the division as multiplication by the reciprocal first.
除法时忘记翻转第二个分式：这是非常常见的错误。始终先将除法改写为乘以倒数。
Incorrect factorisation: A sign error in factorising, especially difference of squares or quadratics, will make the whole answer wrong. Check your factorisation by expanding.
因式分解错误：因式分解中的符号错误，尤其是平方差或二次式，会使整个答案出错。通过展开来检查你的因式分解。
Forgetting excluded values: If a denominator becomes zero, the fraction is undefined. In final answers, it is good practice to state which values of x are not allowed.
忘记排除值：如果分母变为零，分式无定义。在最终答案中，最好指出哪些 x 值不允许。

8. Mixed Operations (Multiplication & Division) | 混合运算（乘除）

When a problem combines multiplication and division, treat it as a sequence of steps. Rewrite every division as multiplication by the reciprocal, then carry out one multiplication and cancel systematically.

当一个题目组合了乘法和除法时，将其视为一系列步骤。将每个除法改写为乘以倒数，然后进行相乘并系统地约分。

Example: Simplify (x/2) ÷ (3/4) × (x/5).

示例：化简 (x/2) ÷ (3/4) × (x/5)。

Step 1: Flip only the fraction after the division sign: (x/2) × (4/3) × (x/5).

第一步：只翻转除号后面的分式：(x/2) × (4/3) × (x/5)。

Step 2: Multiply numerators and denominators: (x × 4 × x) / (2 × 3 × 5) = (4x²)/30 = (2x²)/15.

第二步：分子分母相乘：(x × 4 × x) / (2 × 3 × 5) = (4x²)/30 = (2x²)/15。

Remember to cancel if possible. Here 4/30 simplifies to 2/15.

记住尽可能约分。这里 4/30 化简为 2/15。

For more complex mixed operations with polynomials, factorise first, then flip all divisors, then cancel. For example:

对于更复杂的含多项式混合运算，先因式分解，再翻转所有除数，然后约分。例如：

(x² − 4)/x ÷ (x − 2)/(x + 2) × (x + 2)/x

Factorise x² − 4 = (x − 2)(x + 2). Flip the divisor: (x − 2)(x + 2)/x × (x + 2)/(x − 2) × (x + 2)/x. Cancelling (x − 2) and (x + 2) leaves (x + 2)(x + 2)/(x × x) = (x + 2)² / x².

因式分解 x² − 4 = (x − 2)(x + 2)。翻转除数：(x − 2)(x + 2)/x × (x + 2)/(x − 2) × (x + 2)/x。约去 (x − 2) 和 (x + 2)，剩下 (x + 2)(x + 2)/(x × x) = (x + 2)² / x²。

9. Practice Questions with Solutions | 练习题与解答

Mastery requires practice. Here are carefully selected exam-style questions with full solutions.

掌握需要练习。以下是精心挑选的考试风格题目及完整解答。

Question 1: Simplify 3a/4b × 8b²/9a.

题目 1：化简 3a/4b × 8b²/9a。

Solution: Cancel: 3 and 9 → 1 and 3; 4 and 8 → 1 and 2; a/a = 1; b²/b = b. Thus (1 × 1 × b)/(1 × 3 × 1) = b/3.

解答：约分：3 与 9 → 1 与 3；4 与 8 → 1 与 2；a/a = 1；b²/b = b。因此 (1 × 1 × b)/(1 × 3 × 1) = b/3。

Question 2: Simplify (2x + 4)/(x² − 1) ÷ (x + 2)/(x − 1).

题目 2：化简 (2x + 4)/(x² − 1) ÷ (x + 2)/(x − 1)。

Solution: Factorise: 2x + 4 = 2(x + 2); x² − 1 = (x − 1)(x + 1). Flip the divisor: 2(x + 2)/((x − 1)(x + 1)) × (x − 1)/(x + 2). Cancel (x + 2) and (x − 1). Result: 2/(x + 1).

解答：因式分解：2x + 4 = 2(x + 2)；x² − 1 = (x − 1)(x + 1)。翻转除数：2(x + 2)/((x − 1)(x + 1)) × (x − 1)/(x + 2)。约去 (x + 2) 和 (x − 1)。结果：2/(x + 1)。

Question 3: Simplify (x² + 3x + 2)/(x² + 4x + 3) × (x² + 5x + 6)/(x² + 2x).

题目 3：化简 (x² + 3x + 2)/(x² + 4x + 3) × (x² + 5x + 6)/(x² + 2x)。

Solution: Factorise each quadratic: x² + 3x + 2 = (x + 1)(x + 2); x² + 4x + 3 = (x + 1)(x + 3); x² + 5x + 6 = (x + 2)(x + 3); x² + 2x = x(x + 2). Now multiply and cancel:

解答：因式分解每个二次式：x² + 3x + 2 = (x + 1)(x + 2)；x² + 4x + 3 = (x + 1)(x + 3)；x² + 5x + 6 = (x + 2)(x + 3)；x² + 2x = x(x + 2)。现在相乘并约分：

((x+1)(x+2))/((x+1)(x+3)) × ((x+2)(x+3))/(x(x+2)) = (x+2)/x

After cancelling (x + 1), (x + 3), and one (x + 2), we get (x + 2)/x.

约去 (x + 1)、(x + 3) 和一个 (x + 2) 后，得到 (x + 2)/x。

10. Exam Tips | 考试技巧

To maximise your score in the IGCSE Edexcel exam, keep the following strategies in mind.

要在 IGCSE Edexcel 考试中最大化得分，请牢记以下策略。

Factorise first, cancel later: Always factorise all polynomials before attempting to cancel. This is the single most important habit.
先因式分解，再约分：在尝试约分之前，务必先将所有多项式因式分解。这是最重要的习惯。
Write down every step: Even if you can do the simplification mentally, writing stages helps you avoid sign errors and allows the examiner to award method marks.
写出每一步：即使你可以心算化简，写出步骤有助于避免符号错误，也让阅卷官能够给方法分。
State excluded values when required: If a question asks for restrictions, list values that make any original denominator zero. For example, in 2/(x+1), x ≠ −1.
在必要时写出排除值：如果题目要求限制条件，列出使任何原分母为零的值。例如，在 2/(x+1) 中，x ≠ −1。
Check by substituting a simple value: Choose a number like x = 2 (that doesn’t make denominators zero), substitute into the original expression and into your simplified answer. If both give the same value, you are very likely correct.
用简单数值检验：选一个不会使分母为零的数，如 x = 2，分别代入原表达式和化简后的答案。若两者结果相同，你很可能做对了。
Be careful with signs: A common factor like (x − 2) and (2 − x) are not the same but are related: 2 − x = −(x − 2). You can factor out −1 to make cancellation possible.
注意符号：公因子 (x − 2) 与 (2 − x) 并不相同，但有关联：2 − x = −(x − 2)。你可以提出 −1 使其能够约分。
Don’t rush: Most errors occur when students skip cancellation steps and try to handle everything at once. Slow and systematic work earns marks.
不要急躁：大多数错误发生在学生跳过约分步骤、试图一步到位时。慢而系统的工作才能拿分。

With consistent practice, multiplying and dividing algebraic fractions becomes a routine process. Remember the golden rules: factorise, flip, cancel, and verify.

通过持续练习，代数分式的乘除会变成一个常规过程。记住黄金法则：因式分解、翻转、约分、验证。

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Multiplying and Dividing Algebraic Fractions | 代数分式的乘除