📚 Chapter Review 1: Algebraic Expressions (Edexcel A-Level Pure Maths) | 章节复习 1：代数表达式（Edexcel A-Level 纯数）
Welcome to Chapter Review 1 for Edexcel A-Level Pure Mathematics. This review consolidates the foundation skills covered in the opening chapter – algebraic manipulation, laws of indices, surds, quadratics and algebraic fractions. Mastering these techniques is essential because they underpin almost every topic that follows, from calculus to trigonometry. Work through each section carefully, check the worked examples and test your understanding with the self-check exercises.
欢迎来到 Edexcel A-Level 纯数第一章复习。本章巩固开篇所学的代数基础技能——代数运算、指数定律、无理根式、二次表达式与代数分式。熟练掌握这些技巧至关重要，因为它们几乎支撑着后续所有专题，从微积分到三角学。请仔细阅读每个小节，研读例题，并通过自测练习检验理解程度。
1. Laws of Indices | 指数定律
Indices (or exponents) control how we multiply and divide powers. For any non‑ aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. Fractional indices link roots and powers: a¹/ⁿ = ⁿ√a, and aᵐ/ⁿ = ⁿ√aᵐ = (ⁿ√a)ᵐ.
指数（幂）控制着幂的乘除运算。对于任意非零底数 a 和整数 m、n，核心法则是：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。分数指数连接根式与幂：a¹/ⁿ = ⁿ√a，aᵐ/ⁿ = ⁿ√aᵐ = (ⁿ√a)ᵐ。
Example: Simplify (2x²y⁻³)³ ÷ (4x⁻¹y)².
ficients: 8/16 = 1/2. For x: x⁶⁻⁽⁻²⁾ = x⁸. For y: y⁻⁹⁻² = y⁻¹¹ = 1/y¹¹. Result: x⁸/(2y¹¹).
先展开：2³x⁶y⁻⁹ ÷ (4²x⁻²y²) = 8x⁶y⁻⁹ ÷ (16x⁻²y²)。系数相除：8/16 = 1/2。x 部分：x⁶⁻⁽⁻²⁾ = x⁸。y 部分：y⁻⁹⁻² = y⁻¹¹ = 1/y¹¹。结果为 x⁸/(2y¹¹)。
2. Expanding and Simplifying | 展开与化简
) method. Always collect like terms afterwards. Watch for hidden signs: (a − b)(c + d) = a⋅c + a⋅d − b⋅c − b⋅d.
展开括号是指将括号外的项与括号内的每一项相乘。对于两个二项式，使用 FOIL 法则（首、外、内、尾）。最后一定要合并同类项。注意符号陷阱：(a − b)(c + d) = a⋅c + a⋅d − b⋅c − b⋅d。
Worked example: Expand and simplify (3x − 2)(x² + 5x − 1). Multiply 3x by each term: 3x³ + 15x² − 3x. Then −2 by each term: −2x² − 10x + 2. Combine: 3x³ + 13x² − 13x + 2.
例题：展开并化简 (3x − 2)(x² + 5x − 1)。3x 乘各项：3x³ + 15x² − 3x。−2 乘各项：−2x² − 10x + 2。合并：3x³ + 13x² − 13x + 2。
x + 4)² = (x + 4)(x + 4) = x² + 8x + 16. The common mistake x² + 16 loses the middle term.
平方括号时，要写成乘法形式：(x + 4)² = (x + 4)(x + 4) = x² + 8x + 16。常见错误是写为 x² + 16，漏掉了中间项。
3. Factorising by Common Factors | 提取公因式
Factorising is the reverse of expanding. Always look for a common factor first – f all terms. For example, 6x³ + 9x² − 12x = 3x(2x² + 3x − 4). Check by expanding mentally.
因式分解是展开的逆运算。永远先寻找各项的公因式——最高公因式 (HCF)。例如，6x³ + 9x² − 12x = 3x(2x² + 3x − 4)。在心算中展开验证一下。
= (y + 3)(x + 2). This technique is invaluable in later topics such as partial fractions.
如果表达式有四项，可尝试分组分解：xy + 3x + 2y + 6 = x(y + 3) + 2(y + 3) = (y + 3)(x + 2)。这一技巧对后续部分分式等专题非常有用。
4. Factorising Quadratics | 二次三项式的因式分解
q = b and pq = c. When the coefficient of x² is not 1, for instance ax² + bx + c, use the 'ac method' : find two numbers that multiply to ac and add to b, split the middle term, then factor by grouping.
形如 x² + bx + c 的二次式可分解为 (x + p)(x + q)，其中 p + q = b，pq = c。当 x² 的系数不为 1 时，例如 ax² + bx + c，可用 'ac 方法'：找两个数，其积为 ac、和为 b，拆分中间项，然后分组分解。
+11: +15 and −4. Rewrite: 6x² + 15x − 4x − 10 = 3x(2x + 5) − 2(2x + 5) = (2x + 5)(3x − 2).
例题：分解 6x² + 11x − 10。ac = −60，和为 +11 的数是 +15 与 −4。重写：6x² + 15x − 4x − 10 = 3x(2x + 5) − 2(2x + 5) = (2x + 5)(3x − 2)。
Always check your factorisation by expanding quickly. In Edexcel exams, marks are often awarded even if only the correct factors are shown.
务必快速展开验证分解结果。在 Edexcel 考试中，仅写出正确因式通常即能得分。
5. Difference of Two Squares | 平方差公式
The identity a² − b² = (a − b)(a + b) is one of the most repeated patterns in A‑ se with fractions and multiple variables: 9x² − 25y² = (3x)² − (5y)² = (3x − 5y)(3x + 5y).
恒等式 a² − b² = (a − b)(a + b) 是 A-Level 数学中最常出现的模式之一。它对任意代数项均成立，包括含分数及多变量的式子：9x² − 25y² = (3x)² − (5y)² = (3x − 5y)(3x + 5y)。
1). You may need to take out a common factor first: 2x² − 8 = 2(x² − 4) = 2(x − 2)(x + 2).
隐形的平方差经常出现：4x² − 1 = (2x − 1)(2x + 1)。有时需要先提取公因式：2x² − 8 = 2(x² − 4) = 2(x − 2)(x + 2)。
6. Completing the Square | 配方法（完成平方）
Completing the square rewrites a quadratic ax² + bx + c into the form a(x + p)² + q. It reveals the vertex of a parabola at (−p, q) and is essential for deriving the quadratic formula and for integrating rational functions.
配方法将二次式 ax² + bx + c 转化为 a(x + p)² + q 的形式。它能直接给出抛物线的顶点坐标 (−p, q)，并且对推导求根公式、积分有理函数至关重要。
e constant. Example: x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4. So vertex is (−3, −4).
对于 x² + bx + c 的步骤：取 x 系数一半得 p，平方，再调整常数项。例如：x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4。顶点为 (−3, −4)。
factor it out first: 2x² + 8x + 3 = 2[x² + 4x] + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5.
若首项系数 a ≠ 1，先提取公因子：2x² + 8x + 3 = 2[x² + 4x] + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5。
7. The Quadratic Formula and Discriminant | 二次求根公式与判别式
The solutions of ax² + bx + c = 0 are given by the quadratic formula:
x = [−b ± √b² − 4ac] / (2a)
< 0 gives no real roots. In Edexcel, you must link the sign of the discriminant to the number of intersections with the x‑axis.
二次方程 ax² + bx + c = 0 的解由求根公式给出：x = [−b ± √b² − 4ac] / (2a)。根号下的 b² − 4ac 叫做判别式 (Δ)。Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 没有实根。在 Edexcel 考试中，你必须将判别式的符号与图像与 x 轴的交点个数联系起来。
Find k for which x² + kx + 9 = 0 has exactly one real root. Set Δ = k² − 36 = 0 → k = ±6.
例题：求 k 使方程 x² + kx + 9 = 0 只有一个实根。设 Δ = k² − 36 = 0 → k = ±6。
8. Surds and Rationalising Denominators | 无理根式与分母有理化
Surds are irrational roots like √2, √3. You can simplify them using the rule √ab = √a × √b. Always express surds in simplest form: √50 = √25×2 = 5√2. To rationalise a denominator with a single surd, multiply numerator and denominator by that surd: 3/√5 = (3√5)/5.
无理根式（根号数）如 √2、√3。可用法则 √ab = √a × √b 化简。根式要化成最简形式：√50 = √25×2 = 5√2。对于分母为单一根式的有理化，将分子分母同乘该根式：3/√5 = (3√5)/5。
For a denominator like a + b√c, multiply by the conjugate a − b√c. Example: Rationalise 2/(3 + √7). Multiply top and bottom by 3 − √7: 2(3 − √7) / (9 − 7) = (6 − 2√7)/2 = 3 − √7.
若分母形如 a + b√c，乘以其共轭根式 a − b√c。例题：化简 2/(3 + √7)。分子分母同乘 3 − √7：2(3 − √7) / (9 − 7) = (6 − 2√7)/2 = 3 − √7。
You'll frequently encounter surds in coordinate geometry and trigonometry; keeping answers in exact form is an expected skill.
在解析几何和三角学中会频繁遇到根式；用精确值表示答案是必备技能。
9. Working with Algebraic Fractions | 代数分式的运算
Simplifying algebraic fractions requires factorising numerators and denominators first, then cancelling common factors. Always state restrictions (values that make a denominator zero). For addition or subtraction, find a common denominator, combine, and simplify.
化简代数分式首先要对分子分母进行因式分解，再约去公因式。务必注明分母为零的情况（限制条件）。做加减法时，先通分、合并，再化简。
Example: Simplify (x² − 9)/(x² + x − 12). Factorise: (x − 3)(x + 3) / [(x + 4)(x − 3)]. Cancel (x − 3): (x + 3)/(x + 4), x ≠ 3, x ≠ −4.
例题：化简 (x² − 9)/(x² + x − 12)。分解：(x − 3)(x + 3) / [(x + 4)(x − 3)]。约去 (x − 3)：得到 (x + 3)/(x + 4)，x ≠ 3, x ≠ −4。
When adding, treat the numerator carefully. Example: 1/(x + 2) + 3/(x − 1) = [1⋅(x − 1) + 3⋅(x + 2)] / [(x + 2)(x − 1)] = (4x + 5)/[(x + 2)(x − 1)].
做加法时，分子要仔细处理。例题：1/(x + 2) + 3/(x − 1) = [1⋅(x − 1) + 3⋅(x + 2)] / [(x + 2)(x − 1)] = (4x + 5)/[(x + 2)(x − 1)]。
10. Common Pitfalls and Exam Success Tips | 常见错误与考试取胜技巧
Misapplication of index laws, especially a − aⁿ is not aⁿ⁻¹, and forgetting to apply powers to coefficients when expanding brackets are top errors. Also, always check that your factorisation genuinely multiplies back; examiners will trap you with sign errors. Rushing through 'easy' surds often costs marks – write each step.
指数定律的错误应用是首要失分点，例如 a − aⁿ 不等于 aⁿ⁻¹；展开括号时忘记将系数也乘方也是常见错误。另外，一定要检查分解后的因式相乘是否还原；考官会用符号错误设置陷阱。匆忙处理“简单”根式常导致失分——务必将每一步都写出来。
onfirm discriminant decisions. On algebraic fractions, never cancel terms unless the whole expression is factorised. Practise completing the square until it becomes automatic, because it appears heavily in Year 2 calculus.
使用求根公式时，符号括号要一丝不苟：当 b 为负数时，−(−5) = +5。快速画图验证判别式的判断。处理代数分式时，除非整个表达式已完全分解，否则绝不能直接约去项。持续练习配方法，直到变得自动，因为它在第二年的微积分中频繁出现。
| Skill Area | Quick Self‑Check |
|---|---|
| Indices | Simplify (2x³y⁻⁴)² ÷ (4x⁻²y)³ and express with positive indices. |
| Quadratics | Solve 2x² − 5x − 3 = 0 by factorising, completing square and formula. |
| Surds | Express (√3 + 2)/(√3 − 1) in the form a + b√3. |
Consistent, deliberate practice of these fundamentals will save time in the exam and build a solid base for the tougher topics ahead.
有针对性地、持续地练习这些基础内容，不仅能节省考试时间，还能为后续更难的知识点打下牢固基础。
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