Algebraic Manipulation | 代数式的运算与化简

📚 Algebraic Manipulation | 代数式的运算与化简

Algebraic manipulation is a core skill in the IGCSE Mathematics syllabus. It underpins everything from simple equations to advanced graph sketching. This revision guide breaks down the essential techniques into short, focused sections, each with a clear example and an explanation of common pitfalls.

代数式的运算与化简是 IGCSE 数学大纲中的核心技能，也是解决方程、画图乃至进阶数学问题的基础。这本复习指南将关键技巧拆解成简短而聚焦的小节，每节都配有清晰例题和常见错误提醒。

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1. Collecting Like Terms | 合并同类项

Like terms have exactly the same variable parts raised to the same powers. For example, (3x) and (-5x) are like terms, while (3x) and (3x^2) are not. To simplify, add or subtract their coefficients.

同类项是指变量部分完全相同且幂次也相同的项。例如 (3x) 与 (-5x) 是同类项，而 (3x) 与 (3x^2) 则不是。化简时只需对系数进行加减。

2x² + 3x − x² + 5x = x² + 8x

Be careful with signs: subtracting a negative term changes the sign. Always rearrange so that the variable part is written with its sign attached.

特别注意符号：减去一个负数等价于加上它的相反数。建议移动项时连同符号一起处理。

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2. Expanding Brackets | 去括号

Use the distributive law: a(b + c) = ab + ac. When a bracket follows another bracket, multiply each term of the first bracket by each term of the second.

使用乘法分配律：a(b + c) = ab + ac。当两个括号相乘时，用第一个括号的每一项分别乘以第二个括号的每一项。

(x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6

Check: after expanding, count the number of terms. Two linear brackets multiplied should give four terms before simplification. If you only get three, you probably missed one.

检查：展开后数一数项数。两个一次括号相乘应先得到四项，再合并同类项；若只得到三项，多半是漏乘了一项。

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3. Factorising by Common Factor | 提取公因式

Factorising is the reverse of expanding. Look for the greatest common factor (GCF) of all terms and place it outside the bracket.

因式分解是去括号的逆运算。先找出所有项的最大公因式，将其提到括号外面。

6x³ + 9x² = 3x²(2x + 3)

Always check that the remaining bracket has no common factor left. In this case, (2x) and (3) have no common factor apart from 1.

务必检查括号内是否还能继续提取公因式。此例中 (2x) 与 (3) 除了 1 以外没有任何公因数。

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4. Factorising Quadratics | 因式分解二次三项式

For a quadratic of the form x² + bx + c, find two numbers that multiply to give c and add to give b. Those numbers replace the middle term before grouping.

对于形如 x² + bx + c 的二次三项式，需找到两个数，使它们的乘积为 c，和为 b。然后将中间项拆分，再用分组法因式分解。

x² − 5x + 6 = (x − 2)(x − 3)

When the x² coefficient is not 1, use the ac method: multiply the constant by the leading coefficient, then factor that product. Practice with several examples to become fluent.

当 x² 的系数不为 1 时，使用 ac 法：先把二次项系数与常数项相乘，再对所得积进行因数分解。多练习几个例子就能熟练掌握。

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5. Solving Linear Equations | 解一元一次方程

To solve a linear equation, isolate the variable on one side using inverse operations. Perform the same operation on both sides to maintain balance.

解一元一次方程时，通过逆运算将变量单独放在等号一侧。必须对等号两边同时进行相同的运算，以保持方程平衡。

4x − 7 = 2x + 5 → 2x = 12 → x = 6

If the variable appears on both sides, first bring them together on the left by subtracting the smaller coefficient. Then continue with addition or subtraction.

如果变量同时出现在等号两边，先把较小的变量项移到左边，再通过加减常数项继续求解。

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6. Simultaneous Equations | 联立方程组

Two linear equations in two unknowns can be solved by elimination or substitution. Elimination is often quickest when the coefficients of one variable are made equal.

两个二元一次方程可以用消元法或代入法求解。当某个变量的系数能设法变成相等时，消元法通常最快。

2x + y = 7 → 6x + 3y = 21
x − 3y = 14 → x − 3y = 14
7x = 35 → x = 5, y = −3

Check your solution by substituting both values into the original equations. A correct pair must satisfy both equations simultaneously.

将解代回原方程进行验算。正确的解必须同时满足两个方程。

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7. Indices and Powers | 指数法则

The laws of indices are essential shortcuts: (a^m times a^n = a^{m+n}), (a^m ÷ a^n = a^{m-n}), and ((a^m)^n = a^{mn}).

指数的运算法则非常重要：(a^m times a^n = a^{m+n})，(a^m ÷ a^n = a^{m-n})，以及 ((a^m)^n = a^{mn})。

x⁴ × x⁻² = x²

Remember that (x^0 = 1) for any nonzero x, and negative powers mean reciprocals: (x^{-3} = frac{1}{x^3}). Use these rules to simplify algebraic expressions efficiently.

记住：任何非零数的 0 次方等于 1；负指数表示倒数：(x^{-3} = frac{1}{x^3})。熟练运用这些法则能高效化简代数式。

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8. Algebraic Fractions | 代数分式

Treat algebraic fractions like numerical ones. Factorise numerators and denominators first, then cancel common factors. Never cancel terms inside a sum or difference.

代数分式的运算与数字分式相同。先对分子分母因式分解，再约去公因子。千万不能约分和式或差式中的项。

(x² − 9) / (x² + 3x) = (x − 3)(x + 3) / [x(x + 3)] = (x − 3) / x

When adding fractions, find a common denominator by taking the product of the denominators if no simpler option exists. Simplify the final answer as far as possible.

分式加减时，如果没有更简单的公分母，可将两个分母相乘作为公分母。最后结果要化简到最简形式。

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9. Rearranging Formulas | 公式变形

Making one variable the subject of a formula is similar to solving an equation. Treat the target variable as the unknown and apply inverse operations to everything else.

把公式中的某个变量表示成其他变量的函数，类似于解方程。把目标变量视为未知数，对其余部分进行逆运算。

A = ½(a + b)h → h = 2A / (a + b)

Be systematic: remove fractions first, then brackets, then square roots. Keep the subject variable positive if possible, and never lose a negative sign when dividing.

按顺序操作：先去分数，再去括号，最后处理平方根。尽量保持主变量取正数，在除法时不要丢掉负号。

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10. Problem Solving with Algebra | 代数应用题

Many IGCSE paper 2 questions present a scenario in words. Define a variable for the unknown, translate the sentences into an equation, solve it, and re-read the question to give an answer in context.

许多 IGCSE 试卷二题目以文字描述情境。先为未知量设定变量，将文字翻译成方程，求解后再回到原题，用带单位的语句作答。

“A number is doubled then 5 is added, giving 13” → 2x + 5 = 13 → x = 4

Equations from geometry may involve perimeter, area or angles. Drawing a sketch helps identify the correct algebraic expression before writing the equation.

几何应用题常涉及周长、面积或角度。先画草图，可以帮助你在列方程前写出正确的代数表达式。

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Above all, algebraic manipulation improves with regular practice. Check each step against the laws of arithmetic, and always verify your final answer whenever possible.

总而言之，代数式的运算与化简需要持之以恒的练习。每一步都要对照算术法则自我检查，并且尽可能验算最终答案。

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