📚 Core Pure Mathematics: Algebraic Methods, Graphs and Circles | 核心纯数学：代数方法、图像与圆
This revision unit covers the core pure mathematics techniques found in a typical A-level programme, including algebraic division, polynomial methods, graph transformations, coordinate geometry of circles, and the links between algebra and curve sketching.
本复习单元涵盖 A-level 课程中常见的核心纯数学方法，包括代数除法、多项式方法、图像变换、圆的坐标几何，以及代数与曲线作图之间的联系。
1. Algebraic Division and the Factor Theorem | 代数除法与因式定理
t is written as f(x) = (x − a)Q(x) + R, where Q(x) is the quotient and R is the remainder.
多项式长除法用于将一个三次或更高次多项式除以一个线性因式（如 x − a）。结果写作 f(x) = (x − a)Q(x) + R，其中 Q(x) 是商，R 是余数。
. The factor theorem is a special case: x − a is a factor of f(x) if and only if f(a) = 0.
如果除式是 x − a，余数可以通过计算 f(a) 快速求得。这就是余数定理。因式定理是特殊情况：当且仅当 f(a) = 0 时，x − a 是 f(x) 的因式。
f(x) = (x − a)Q(x) + R
- − 5 = −2. Therefore the remainder when dividing by x − 1 is −2, and x − 1 is not a factor.
- 例：对于 f(x) = 2x³ − 3x² + 4x − 5，计算 f(1) = 2 − 3 + 4 − 5 = −2。因此除以 x − 1 的余数为 −2，x − 1 不是因式。
- ±2, ±3. Once one factor is found, use division or inspection to find the quadratic factor.
- 要对三次式进行因式分解，可尝试小整数值，如 x = ±1、±2、±3。找到一个因式后，用除法或观察法求出二次因式。
2. Polynomial Identities and Remainder Theorem | 多项式恒等式与余数定理
Two polynomials are identically equal if coefficients of corresponding powers of x are equal. This idea is used to find unknown constants in an identity.
两个多项式恒等，当且仅当 x 的同次幂系数相等。这个思想用于求恒等式中的未知常数。
For example, if ax² + bx + c ≡ f(2) = 0, then x − 2 is a factor; if f(−3) = 5, the remainder when dividing by x + 3 is 5.
例如，若 ax² + bx + c ≡ 3x² − 5x + 2，则比较系数可得 a = 3，b = −5，c = 2。余数定理也可逆用：如果 f(2) = 0，则 x − 2 是一个因式；如果 f(−3) = 5，则除以 x + 3 的余数为 5。
If f(x) ≡ (x − a)Q(x) + R, then R = f(a)
This compact statement links division, evaluation, and factor testing in one line. When an identity contains fractions, multiply through by the common denominator to produce a polynomial identity, then equate coefficients.
这一简洁表达式将除法、代入求值和因式检验联系在一起。当恒等式中含有分式时，可先乘以公分母得到多项式恒等式，再比较系数。
3. Graphs and Transformations | 函数图像与变换
can be described by changes inside or outside the function bracket. Understanding the direction and scale of each transformation is essential for sketching transformed graphs quickly.
函数图像 y = f(x) 的变换可以通过括号内外的变化来描述。理解每种变换的方向和比例，对于快速画出变换后的图像至关重要。
- y = f(x) + a: translation upwards by a units.
- y = f(x) + a：向上平移 a 个单位。
- y = f(x + a): translation left by a units when a > 0.
- y = f(x + a)：当 a > 0 时向左平移 a 个单位。
- y = af(x): vertical stretch by scale factor a.
- y = af(x)：竖直方向拉伸，比例因子为 a。
- y = f(ax): horizontal stretch by scale factor 1/a.
- y = f(ax)：水平方向拉伸，比例因子为 1/a。
- y = −f(x): reflection in the x-axis.
- y = −f(x)：关于 x 轴反射。
- y = f(−x): reflection in the y-axis.
- y = f(−x)：关于 y 轴反射。
When combining transformations, apply horizontal changes first, then vertical changes, unless brackets indicate otherwise.
组合变换时，通常先进行水平变换，再进行竖直变换，除非括号另有说明。
4. Sketching Quadratics and Cubics | 二次与三次函数草图
A quadratic can be sketched by completing the square to find its vertex and by factorising or using the discriminant to find its x-intercepts. The completed-square form also shows the axis of symmetry.
二次函数的草图可通过完成平方求顶点，并通过因式分解或判别式求 x 轴截距。完全平方式还显示了对称轴。
y = a(x − h)² + k
The vertex is (h, k). The graph is a parabola that opens upwards if a > 0 and downwards if a < 0.
顶点为 (h, k)。图像为抛物线，若 a > 0 开口向上，若 a < 0 开口向下。
Δ = b² − 4ac
ated real root. If Δ < 0, there are no real roots, so the graph does not cross the x-axis.
如果 Δ > 0，有两个不同实根；如果 Δ = 0，有一个重根；如果 Δ < 0，无实根，图像不与 x 轴相交。
he end behaviour: for positive leading coefficient, y → −∞ as x → −∞ and y → +∞ as x → +∞.
对于三次函数，完全因式分解可求根及其重数。重根处图像与 x 轴相切，单根处图像穿过 x 轴。首项系数决定尾端行为：首项系数为正时，当 x → −∞ 时 y → −∞，当 x → +∞ 时 y → +∞。
5. Straight Line Graphs and Modelling | 直线图像与建模
equation of a line can be written as y − y₁ = m(x − x₁), which is the point-gradient form.
经过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂
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