Algebraic Expressions | 代数表达式

📚 Algebraic Expressions | 代数表达式

Algebraic expressions form the very core of A-Level Mathematics. Being able to manipulate, simplify, factorise, expand, and rewrite expressions in different forms is essential for success in topics ranging from coordinate geometry to calculus. This comprehensive guide covers the key algebraic techniques required by the Edexcel specification, including laws of indices, expansion and factorisation, quadratics, completing the square, algebraic fractions, partial fractions, binomial expansions, polynomial division, and algebraic proof. Each section is designed to strengthen both your conceptual understanding and your fluency in algebraic manipulation, preparing you for exam-style problems.

代数表达式是 A-Level 数学的核心基础。无论是对坐标几何还是微积分的学习，熟练地变形、化简、因式分解、展开以及将表达式改写成不同形式都是必不可少的技能。本指南全面覆盖了 Edexcel 考纲所要求的主要代数技巧，包括指数定律、展开与因式分解、二次式、配方法、代数分式、部分分式、二项式展开、多项式除法以及代数证明。每一个小节都旨在加深你对概念的理解并提高代数运算的熟练度，从而帮你做好应对考试题型的准备。

1. Laws of Indices | 指数定律

The laws of indices allow us to simplify expressions involving powers efficiently. When working with aᵐ and aⁿ, where a is a non-zero base and m, n are real numbers, the following rules apply.

指数定律让我们能够高效地化简含有幂的表达式。在处理 aᵐ 和 aⁿ（其中 a 是非零底数，m、n 为实数）时，以下法则适用。

aᵐ × aⁿ = aᵐ⁺ⁿ

The product rule states that when multiplying powers with the same base, we add the exponents.

乘法法则：同底数幂相乘，指数相加。

aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)

The quotient rule tells us that when dividing powers with the same base, we subtract the exponents.

除法法则：同底数幂相除，指数相减（a ≠ 0）。

(aᵐ)ⁿ = aᵐⁿ

The power of a power rule requires us to multiply the exponents when a power is raised to another exponent.

幂的乘方法则：幂的乘方时，指数相乘。

a⁰ = 1 (a ≠ 0)

Any non-zero number raised to the power of zero equals 1. This follows logically from the quotient rule.

任何非零数的零次幂都等于 1。这可以从除法法则合理推出。

a⁻ⁿ = 1 / aⁿ

A negative exponent indicates the reciprocal of the positive power of the base.

负指数表示该底数的正指数幂的倒数。

a¹/ⁿ = ⁿ√a and aᵐ/ⁿ = ⁿ√aᵐ

Fractional exponents correspond to roots: the denominator of the fraction gives the root, and the numerator gives the power.

分数指数对应于根式：分母表示开方次数，分子表示乘方次数。

le: Simplify (8x³y⁻²)²/³. First apply the outer exponent: (8)²/³ × (x³)²/³ × (y⁻²)²/³ = (³√8)² × x² × y⁻⁴/³ = 4x² / y⁴/³. These rules are fundamental and must be second nature.

示例：化简 (8x³y⁻²)²/³。先应用外层的指数：(8)²/³ × (x³)²/³ × (y⁻²)²/³ = (∛8)² × x² × y⁻⁴/³ = 4x² / y⁴/³。这些基本法则必须熟练到不假思索的程度。

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2. Expanding Brackets | 展开括号

c. When dealing with two binomials, we apply the FOIL method or systematic multiplication.

展开括号就是将括号内的表达式乘出来的过程，依据的是分配律：a(b + c) = ab + ac。对于两个二项式相乘，我们可以使用首外内尾（FOIL）方法或系统地进行逐项相乘。

(x + a)(x + b) = x² + (a + b)x + ab

quadratic expression. Pay attention to signs; for instance, (x - 3)(x + 5) = x² + 2x - 15.

两个一次二项式的乘积会产生一个二次表达式。注意符号；例如 (x - 3)(x + 5) = x² + 2x - 15。

(a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b²

Perfect square expansions are extremely common. Recognising the pattern allows for quick expansion and is also vital for factorisation and completing the square.

完全平方展开极为常见。识别这一模式不仅能快速展开，对因式分解和配方法也至关重要。

(a + b)(a - b) = a² - b²

The difference of two squares is another standard form. It emerges frequently when simplifying algebraic fractions or factorising.

平方差公式是另一种标准型。在化简代数分式或因式分解时经常出现。

r instance, expanding (x + 2)(x² - 3x + 1) gives x³ - 3x² + x + 2x² - 6x + 2 = x³ - x² - 5x + 2.

对于含有三个或更多项的情形，要系统地将第一个括号中的每一项与第二个括号中的每一项相乘，然后合并同类项。例如，展开 (x + 2)(x² - 3x + 1) 得到 x³ - 3x² + x + 2x² - 6x + 2 = x³ - x² - 5x + 2。

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3. Factorising | 因式分解

Factorising is the reverse process of expanding; it writes an expression as a product of its factors. The first step is always to look for a common factor in all terms.

因式分解是展开的逆过程，它将一个表达式写成其因式的乘积。第一步始终是检查所有项是否有公因式。

ab + ac = a(b + c)

ften reveals a quadratic or other factorisable form. For example, 6x³ - 9x² = 3x²(2x - 3).

提取公因式可以简化表达式，并能揭示出二次式或其他可分解的形式。例如 6x³ - 9x² = 3x²(2x - 3)。

um to b. In the case where a=1, we need two numbers whose product is c and whose sum is b.

对于 ax² + bx + c 型的二次式，我们需要找到两个数，使之相乘得 ac 且相加得 b。当 a=1 时，就只需找两个数乘积为 c 且和为 b。

x² + 5x + 6 = (x + 2)(x + 3)

This is the most common type. With a leading coefficient other than 1, we often factorise by grouping or trial and error.

这是最常见的类型。当首项系数不为 1 时，我们常用分组分解或试错法进行因式分解。

2x² + 7x + 3 = (2x + 1)(x + 3)

tremely useful, especially when dealing with expressions like 4x² - 25 = (2x - 5)(2x + 5).

识别平方差 a² - b² = (a - b)(a + b) 的模型非常有用，尤其是处理诸如 4x² - 25 = (2x - 5)(2x + 5) 这样的表达式时。

, for instance x³ + x² - x - 1 = x²(x + 1) - 1(x + 1) = (x² - 1)(x + 1) = (x - 1)(x + 1)².

更高级的因式分解可能涉及分组，例如 x³ + x² - x - 1 = x²(x + 1) - 1(x + 1) = (x² - 1)(x + 1) = (x - 1)(x + 1)²。

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4. Quadratic Expressions | 二次表达式

A quadratic expression is any expression of the form ax² + bx + c, where a, b, and c are constants and a ≠ 0. Much of A-Level algebra revolves around manipulating quadratics to solve equations and sketch graphs.

二次表达式是形如 ax² + bx + c 的表达式，其中 a、b、c 是常数且 a ≠ 0。A-Level 数学中有大量代数运算都是围绕二次式展开的，用于求解方程和绘制图像。

Quadratic formula: x = [-b ± √b² - 4ac] / (2a)

ots: two real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.

求根公式可以给出方程 ax² + bx + c = 0 的根。判别式 Δ = b² - 4ac 决定根的性质：若 Δ > 0，有两个不等的实根；若 Δ = 0，有一个重根；若 Δ < 0，没有实根。

Quadratics can be expressed in various forms: standard form ax² + bx + c, factorised form (px + q)(rx + s) when roots are rational, and completed square form a(x + p)² + q which gives the vertex.

二次式可以写成多种形式：标准型 ax² + bx + c、当根为有理数时的因式分解型 (px + q)(rx + s)，以及可给出顶点的配方式 a(x + p)² + q。

Understanding how to move between these forms is a core skill. For example, the expression 2x² - 8x + 6 can be factorised as 2(x - 1)(x - 3) and also written in completed square form as 2(x - 2)² - 2.

熟练掌握这些形式之间的转换是一项核心技能。例如，表达式 2x² - 8x + 6 既可以因式分解为 2(x - 1)(x - 3)，也可以写出配方式 2(x - 2)² - 2。

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5. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x + p)² + q, revealing its minimum or maximum value and the line of symmetry. This technique is essential for solving quadratics, integrating, and finding the vertex of a parabola.

配方法将二次表达式改写为 a(x + p)² + q 的形式，由此能够直接看出其最小值或最大值以及对称轴。这项技巧对求解二次方程、积分以及寻找抛物线的顶点都至关重要。

For a monic quadratic x² + bx, we add and subtract (b/2)² to form (x + b/2)² - (b/2)².

对于首一二次式 x² + bx，我们加上并减去 (b/2)² 来构成 (x + b/2)² - (b/2)²。

x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4

When the coefficient 'a' 8x + 3, we write 2(x² - 4x) + 3 = 2[(x - 2)² - 4] + 3 = 2(x - 2)² - 8 + 3 = 2(x - 2)² - 5.

当系数 a 不为 1 时，需要先从含 x 的项中将其提出来。例如 2x² - 8x + 3，写成 2(x² - 4x) + 3 = 2[(x - 2)² - 4] + 3 = 2(x - 2)² - 8 + 3 = 2(x - 2)² - 5。

oving the constant, and completing the square leads directly to the formula. It is also the method behind finding the turning point of a quadratic graph: the vertex is at (-p, q).

配方法也可用于推导求根公式。从 ax² + bx + c = 0 出发，除以 a，移常数项，配方后可直接推导出求根公式。它也是求二次函数图像拐点的方法：顶点坐标为 (-p, q)。

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6. Algebraic Fractions | 代数分式

Algebraic fractions behave like numerical fractions but involve polynomials. Simplifying them often requires factorisation and cancellation of common factors, keeping an eye on restrictions where the denominator is zero.

代数分式的运算规则与数值分数相似，只是其中包含多项式。化简时通常需要进行因式分解并约去公因式，同时要留意分母为零的约束条件。

(x² - 1)/(x² + x - 2) = (x - 1)(x + 1) / [(x - 1)(x + 2)] = (x + 1)/(x + 2) (x ≠ 1, -2)

bd), and we cancel any common factors before multiplying. Dividing is the same as multiplying by the reciprocal.

代数分式相乘遵循 (a/b) × (c/d) = (ac)/(bd)，在相乘之前先约去公因式会更简便。分式相除等同于乘以除式的倒数。

Adding and subtracting require a common denominator. For fractions with different denominators, find the lowest common multiple (LCM) of the denominators, rewrite each fraction, then combine the numerators.

加减运算需要通分。对于分母不同的分式，要找出各分母的最低公倍式（LCM），将每个分式分别转换成以该公倍式为分母的等价分式，然后再合并分子。

= 3(x - 1)/[(x - 2)(x + 1)]. Often the numerator can be factorised and simplified further.

例子： 1) + 2(x - 2)] / [(x - 2)(x + 1)] = (3x - 3)/[(x - 2)(x + 1)] = 3(x - 1)/[(x - 2)(x + 1)]。通常分子还可以继续因式分解并化简。

Solving equations involving algebraic fractions typically begins by multiplying through by the common denominator to clear the fractions, then solving the resulting polynomial equation, checking for extraneous solutions.

求解含有代数分式的方程通常要先在方程两边同乘公分母以去分母，然后解所得的多项式方程，并检验是否有增根。

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7. Partial Fractions | 部分分式

Partial fractions decompose a complex rational expression into a sum of simpler fractions. This technique is heavily used in integration, binomial expansions, and solving differential equations. The form of decomposition depends on the factors of the denominator.

部分分式将一个复杂的有理式分解为几个简单分式的和。这一技巧在积分、二项式展开以及解微分方程中应用广泛。分解的形式取决于分母因式的类型。

Case 1: Distinct linear factors. If the denominator has factors like (x + a)(x + b), we write the expression as A/(x + a) + B/(x + b) and solve for constants A and B.

情况一：不同的一次因式。如果分母包含 (x + a)(x + b) 这样的因式，我们设原式等于 A/(x + a) + B/(x + b)，再通过解方程确定常数 A 和 B。

(5x - 1)/[(x + 2)(x - 3)] = A/(x + 2) + B/(x - 3) → A = 3, B = 2

Multiplying through by the common denominator and equating coefficients or substituting suitable values of x gives the values of A and B.

两边同时乘以公分母，然后对比系数或代入适当的 x 值，即可求出 A 和 B。

Case 2: Repeated linear factors. For

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