📚 Adding and Subtracting Algebraic Fractions | 代数分式的加减法
Algebraic fractions appear frequently in IGCSE Mathematics. You need to know how to add and subtract them confidently, combining numerators correctly after finding a common denominator. This topic builds on your earlier work with numeric fractions and introduces algebraic manipulation.
代数分式在IGCSE数学中经常出现。你需要熟练掌握它们的加减法，先找到公分母，然后正确合并分子。这个话题建立在你之前对数字分式运算的基础上，并引入了代数运算技巧。
1. Understanding Algebraic Fractions | 理解代数分式
An algebraic fraction is simply a fraction where the numerator and/or denominator contain algebraic expressions, for example, 3/x, (x+1)/(x-2), or 4x/(x²-1).
代数分式就是分子和/或分母中含有代数表达式的分式，例如 3/x、(x+1)/(x-2) 或 4x/(x²-1)。
Just like with ordinary fractions, we can perform addition and subtraction, but we must first ensure the denominators are the same.
就像普通分数一样，我们可以进行加减运算，但必须首先确保分母相同。
2. Simplifying Fractions First | 首先化简分式
Before adding or subtracting, always simplify each algebraic fraction as much as possible. Factorise numerators and denominators, and cancel common factors.
在进行加减法之前，务必将每个代数分式尽可能化简。对分子和分母进行因式分解，并约去公因式。
2, as x²-4 = (x-2)(x+2). This step often makes finding the common denominator much easier.
例如，(x²-4)/(x+2) 可以化简为 x-2，因为 x²-4 = (x-2)(x+2)。这一步往往会使寻找公分母的过程简单很多。
3. Finding the Lowest Common Denominator (LCD) | 寻找最低公分母
The LCD is the smallest expression that each denominator divides into evenly. To find it, factorise each denominator completely, then take the product of the highest powers of all distinct factors.
最低公分母是每个分母都能整除的最小表达式。要找到它，先将每个分母彻底因式分解，然后取所有不同因子的最高次幂的乘积。
mple, if the denominators are x²-1 = (x-1)(x+1) and x²-x = x(x-1), the LCD is x(x-1)(x+1).
例如，如果分母是 x²-1 = (x-1)(x+1) 和 x²-x = x(x-1)，那么 LCD 就是 x(x-1)(x+1)。
4. Equivalent Fractions | 等值分式
Once you have the LCD, rewrite each fraction as an equivalent fraction with the LCD as the denominator. Multiply the numerator and denominator of each fraction by whatever factor is missing from its original denominator.
一旦找到 LCD，就将每个分式改写为以 LCD 为分母的等值分式。用原分母缺少的因式乘以该分式的分子和分母。
For instance, to change 2/(x-1) into a fraction with denominator x(x-1)(x+1), multiply top and bottom by x(x+1), giving 2x(x+1)/[x(x-1)(x+1)].
例如，要将 2/(x-1) 变为分母为 x(x-1)(x+1) 的分式，将分子和分母同乘 x(x+1)，得到 2x(x+1)/[x(x-1)(x+1)]。
5. Adding Fractions – Simple Denominators | 简单分母的加法
When denominators are the same, simply add the numerators and keep the common denominator. If they are different, change to equivalent fractions with the LCD, then add numerators.
当分母相同时，直接将分子相加，并保留公分母。如果分母不同，先将它们化为以 LCD 为分母的等值分式，再相加。
Example: 2/(x+1) + 3/(x+1) = 5/(x+1).
示例：2/(x+1) + 3/(x+1) = 5/(x+1)。
(x+2)/[x(x+2)] and 1/(x+2)= x/[x(x+2)], so sum = (2x+2)/[x(x+2)], which can be simplified.
对于不同分母，例如 1/x + 1/(x+2)，LCD 为 x(x+2)。那么 1/x = (x+2)/[x(x+2)]，1/(x+2)= x/[x(x+2)]，所以它们的和为 (2x+2)/[x(x+2)]，可进一步化简。
6. Subtracting Fractions – Simple Denominators | 简单分母的减法
Subtraction follows the same process, but be very careful with signs. After rewriting with the LCD, subtract the numerators, using brackets to avoid sign errors.
减法遵循相同的步骤，但要格外注意符号。在用 LCD 改写之后，分子相减时要用括号，避免符号错误。
+1)]. Then numerator = 3(x+1) - 2(x-1) = 3x+3 -2x+2 = x+5. So answer = (x+5)/[(x-1)(x+1)].
示例：3/(x-1) - 2/(x+1)。LCD = (x-1)(x+1)。改写：3(x+1)/[(x-1)(x+1)] - 2(x-1)/[(x-1)(x+1)]。然后分子 = 3(x+1) - 2(x-1) = 3x+3 -2x+2 = x+5。因此结果为 (x+5)/[(x-1)(x+1)]。
7. Mixed Expressions and Polynomials | 混合表达式与多项式
Sometimes an expression includes a polynomial term plus or minus an algebraic fraction. Convert the polynomial into a fraction by writing it over 1, then find a common denominator.
有时表达式中会包含一个多项式项加减一个代数分式。将多项式写成分母为 1 的分式，然后寻找公分母。
= x/1. The LCD is (x+3). So x = x(x+3)/(x+3) = (x²+3x)/(x+3). Then sum = (x²+3x+2)/(x+3).
例如，要化简 x + 2/(x+3)，将 x 写作 x/1。LCD 为 (x+3)，所以 x = x(x+3)/(x+3) = (x²+3x)/(x+3)。于是和为 (x²+3x+2)/(x+3)。
8. Common Mistakes to Avoid | 需要避免的常见错误
Mistake 1: Adding denominators directly. Never add denominators when adding fractions; only add numerators once denominators are equal.
错误一：直接相加分母。在相加分式时，绝不可以将分母相加；只有分母统一后才将分子相加。
Mistake 2: Forgetting to multiply the entire numerator when building equivalent fractions. Always distribute the missing factor across the whole numerator.
错误二：构造等值分式时忘记将分子整体相乘。一定要将缺失的因式乘到整个分子上。
Mistake 3: Sign errors in subtraction. Use parentheses before subtracting the second numerator. 3x - (x+2) is not 3x - x+2, but 3x - x - 2.
错误三：减法中的符号错误。在减去第二个分子之前使用括号。3x - (x+2) 不等于 3x - x+2，而应为 3x - x - 2。
Mistake 4: Not simplifying the final answer. Always factorise the result and cancel if possible.
错误四：未化简最终结果。始终对结果进行因式分解，并在可能的情况下约分。
9. Step-by-Step Example 1: Linear Denominators | 步骤示例1：线性分母
Simplify 5/(x-2) + 3/(2x+1).
化简 5/(x-2) + 3/(2x+1)。
Step 1: Denominators are x-2 and 2x+1, which are already factorised linear factors. LCD is (x-2)(2x+1).
步骤1：分母为 x-2 和 2x+1，它们已经是分解后的线性因式。LCD 为 (x-2)(2x+1)。
te equivalent fractions: 5/(x-2) = 5(2x+1)/[(x-2)(2x+1)]; 3/(2x+1) = 3(x-2)/[(x-2)(2x+1)].
步骤2：写出等值分式：5/(x-2) = 5(2x+1)/[(x-2)(2x+1)]；3/(2x+1) = 3(x-2)/[(x-2)(2x+1)]。
Step 3: Add numerators: 5(2x+1) + 3(x-2) = 10x+5 + 3x-6 = 13x - 1.
步骤3：分子相加：5(2x+1) + 3(x-2) = 10x+5 + 3x-6 = 13x - 1。
Step 4: Result is (13x - 1)/[(x-2)(2x+1)]. Check for simplification: numerator cannot be factorised further, so this is final.
步骤4：结果为 (13x - 1)/[(x-2)(2x+1)]。检查是否可以化简：分子无法进一步因式分解，因此这是最终答案。
10. Step-by-Step Example 2: Quadratic Denominators | 步骤示例2：二次分母
Simplify x/(x²-1) - 2/(x+1).
化简 x/(x²-1) - 2/(x+1)。
1: Factorise denominator: x²-1 = (x-1)(x+1). Then expression is x/[(x-1)(x+1)] - 2/(x+1).
步骤1：分解分母：x²-1 = (x-1)(x+1)。于是表达式变为 x/[(x-1)(x+1)] - 2/(x+1)。
Step 2: LCD is (x-1)(x+1). Rewrite second fraction: 2/(x+1) = 2(x-1)/[(x-1)(x+1)].
步骤2：LCD 为 (x-1)(x+1)。改写第二个分式：2/(x+1) = 2(x-1)/[(x-1)(x+1)]。
Step 3: Subtract: numerator = x - 2(x-1) = x - 2x + 2 = 2 - x.
步骤3：相减：分子 = x - 2(x-1) = x - 2x + 2 = 2 - x。
Step 4: Result is (2 - x)/[(x-1)(x+1)], which can be written as -(x-2)/[(x-1)(x+1)]. No further simplification.
步骤4：结果为 (2 - x)/[(x-1)(x+1)]，也可写作 -(x-2)/[(x-1)(x+1)]。无需进一步化简。
11. Simplifying the Result | 化简结果
After adding or subtracting, always check if the resulting fraction can be simplified. Factorise the numerator and denominator completely, then cancel any common factors.
加减运算完成后，务必检查结果分式是否可以化简。将分子和分母彻底因式分解，然后约去任何公因式。
example, a result like (x²+3x+2)/(x+2) factors to (x+1)(x+2)/(x+2) = x+1, provided x ≠ -2.
例如，像 (x²+3x+2)/(x+2) 这样的结果可分解为 (x+1)(x+2)/(x+2) = x+1，前提是 x ≠ -2。
Always state any restrictions on the variable (values that make denominator zero) if the question requires it.
如果题目要求，需注明变量的限制条件（使分母为零的值）。
12. Exam Tips | 考试技巧
1. Write each step clearly, showing how you found the LCD and equivalent fractions. Partial marks are awarded for method.
1. 清晰写出每一步，展示你是如何找到 LCD 和等值分式的。过程有分步得分。
e.g., x=2) into both the original expression and your simplified result, provided it doesn't make any denominator zero.
2. 用一个简单数值（如 x=2）代入原表达式和你的化简结果中进行检验，确保代入值不会使任何分母为零。
3. Be meticulous with signs, especially when subtracting. Use brackets generously.
3. 细心处理符号，特别是在做减法时。多使用括号。
4. Familiarise yourself with factorising quadratics and recognising difference of squares, as this is often required.
4. 熟悉二次式的因式分解和平方差公式，因为这些常常会用到。
5. Practice simplifying complex fractions where the whole expression might be a fraction over a fraction, but these appear less frequently.
5. 多练习复杂分式，尽管“分式套分式”出现频率较低，但掌握基础很重要。
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