Support Pack 3: Contents | 支持包3：内容

📚 Support Pack 3: Contents | 支持包3：内容

In this Support Pack 3, we bring together the essential topics that you need to master for the AQA A-Level Mathematics examination. The pack is designed as a concise content map, helping you navigate through each chapter with confidence. We focus on the core methods, key formulas, and common pitfalls that appear in the exam.

在这个支持包3中，我们汇总了AQA A-Level数学考试中你需要掌握的核心主题。本支持包设计为一张简洁的内容地图，帮助你自信地浏览每一章节。我们聚焦于核心方法、关键公式以及考试中常见的易错点。

1. Contents at a Glance | 内容一览

This Support Pack 3 is structured into three main blocks: Pure Mathematics, Statistics, and Mechanics. Each block is subdivided into focused lessons that match the AQA specification. The list below shows the exact list of topics covered in this pack.

本支持包3分为三大板块：纯数学、统计和力学。每个板块被细分为符合AQA大纲的聚焦课程。下面的列表展示了本支持包所覆盖的完整主题列表。

• Pure Mathematics: Algebra, Trigonometry, Calculus, Numerical Methods.

  纯数学：代数、三角函数、微积分、数值方法。

• Statistics: Probability, Hypothesis Testing.

  统计：概率、假设检验。

• Mechanics: Kinematics, Forces.

  力学：运动学、力。

Each section contains worked examples and practice pointers to help you revise actively.

每个部分都包含例题和练习提示，帮助你主动复习。

2. Polynomial Expressions | 多项式表达式

Polynomials form the foundation of much of A-Level algebra. You need to be able to add, subtract, multiply, and divide polynomials, and to factorise them fully. The factor theorem and remainder theorem are essential tools for solving cubic and quartic equations.

多项式是A-Level代数的基础。你需要能够对多项式进行加、减、乘、除运算，并完全因式分解。因式定理和余数定理是解三次和四次方程的关键工具。

For a polynomial f(x), the remainder when divided by (x – a) is f(a). The factor theorem states that (x – a) is a factor if and only if f(a) = 0.

对于多项式 f(x)，除以 (x – a) 的余数为 f(a)。因式定理指出，当且仅当 f(a) = 0 时，(x – a) 是 f(x) 的因式。

f(x) = (x – a)Q(x) + f(a)

Example: For f(x) = x³ – 2x² – 5x + 6, check that f(1) = 0, so (x – 1) is a factor. Then factorise fully.

例如：对于 f(x) = x³ – 2x² – 5x + 6，验证 f(1) = 0，所以 (x – 1) 是一个因式。然后完全因式分解。

Use the factor theorem to test integer roots, then divide to find the quadratic factor, which can be solved using the quadratic formula.

使用因式定理测试整数根，然后相除得到二次因式，再利用二次公式求解。

3. Quadratic Functions | 二次函数

Quadratics appear throughout A-Level Maths. You must be confident in completing the square, finding roots, and using the discriminant b² – 4ac to determine the nature of roots.

二次函数贯穿A-Level数学。你必须熟练配方、求根，并使用判别式 b² – 4ac 来确定根的性质。

The completed square form is a(x – h)² + k, where (h, k) is the vertex of the parabola.

配方形式为 a(x – h)² + k，其中 (h, k) 是抛物线的顶点。

x = (-b ± √(b² – 4ac)) / 2a

If b² – 4ac > 0, there are two distinct real roots. If b² – 4ac = 0, there is one repeated root. If b² – 4ac < 0, there are no real roots.

如果 b² – 4ac > 0，有两个不等实根；如果 b² – 4ac = 0，有一个重根；如果 b² – 4ac < 0，没有实根。

You should also be able to interpret the discriminant graphically, as the number of times the parabola crosses the x-axis.

你还应该能够在图像上解释判别式，也就是抛物线与x轴相交的次数。

4. Exponentials and Logarithms | 指数与对数

Exponential functions and logarithms are closely related. The natural exponential eˣ and natural logarithm ln(x) are inverses

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