Algebraic Expressions | 代数表达式

📚 Algebraic Expressions | 代数表达式

Algebraic expressions are the foundation of A-Level Mathematics. In Edexcel Pure Mathematics, Topic 1 tests your ability to manipulate indices, surds, brackets, algebraic fractions and polynomials accurately. Confidence here is essential because these skills reappear in calculus, sequences, binomial expansion and coordinate geometry.

代数表达式是 A-Level 数学的基础。在 Edexcel 纯数学中，主题 1 考查你对指数、根式、括号、代数分式和多项式的准确变形能力。这一部分的熟练度至关重要，因为这些技能会在微积分、数列、二项式展开和坐标几何中反复出现。

1. Index Laws | 指数定律

You must be able to apply index laws fluently for integer and rational powers. The key rules are summarised below.

你必须能够熟练运用整数指数和有理指数定律。主要规则总结如下。

aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a, a^(m/n) = ⁿ√(aᵐ)

For example, x³ × x⁵ = x⁸ and (2x²)³ = 8x⁶. Note that a negative power produces a reciprocal, and a fractional power produces a root.

例如，x³ × x⁵ = x⁸，(2x²)³ = 8x⁶。注意负指数表示倒数，分数指数表示开方。

Rule	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	x² × x⁴ = x⁶
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	x⁷ ÷ x³ = x⁴
(aᵐ)ⁿ = aᵐⁿ	(x³)² = x⁶
a⁰ = 1	5⁰ = 1
a⁻ⁿ = 1/aⁿ	x⁻² = 1/x²
a^(1/n) = ⁿ√a	x^(1/2) = √x
a^(m/n) = ⁿ√(aᵐ)	x^(3/2) = √(x³)

2. Expanding Brackets | 展开括号

The distributive law states that a(b + c) = ab + ac. When two binomials are multiplied, every term in the first bracket is multiplied by every term in the second bracket.

分配律指出 a(b + c) = ab + ac。当两个二项式相乘时，第一个括号中的每一项都要与第二个括号中的每一项相乘。

(x + a)(x + b) = x² + (a + b)x + ab

For example, (x + 3)(x − 5) = x² − 2x − 15. The two special expansions you must recognise instantly are:

例如，(x + 3)(x − 5) = x² − 2x − 15。你必须立即识别以下两个特殊展开式：

(a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b²

These identities are particularly useful when factorising difference of squares or completing the square.

这些恒等式在因式分解平方差或进行配方法时特别有用。

3. Factorising Quadratics | 二次式因式分解

Factorising is the reverse of expanding. Start by taking out the highest common factor, then look for the structure of the remaining expression.

因式分解是展开的逆运算。先提取最大公因式，然后观察剩余表达式的结构。

For a monic quadratic x² + bx + c, find two numbers whose product is c and whose sum is b. For example, x² − 7x + 12 = (x − 3)(x − 4) because (−3) × (−4) = 12 and (−3) + (−4) = −7.

对于首一二次式 x² + bx + c，找出两个数，使它们的乘积为 c，和为 b。例如，x² − 7x + 12 = (x − 3)(x − 4)，因为 (−3) × (−4) = 12，且 (−3) + (−4) = −7。

For non-monic quadratics such as 2x² + 7x + 3, split the middle term or use trial and error to obtain (2x + 1)(x + 3).

对于非首一二次式，如 2x² + 7x + 3，可以拆分中间项或用尝试法得到 (2x + 1)(x + 3)。

The difference of two squares is a common target: 9x² − 25 = (3x − 5)(3x + 5).

平方差是常见目标：9x² − 25 = (3x − 5)(3x + 5)。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. It is essential for solving quadratics, finding the vertex, and integrating rational functions.

配方法将二次式改写为 a(x + p)² + q 的形式。它对于解二次方程、求顶点以及积分有理函数都至关重要。

x² + bx = (x + b/2)² − (b/2)²

For example, complete the square for x² − 6x + 5:

例如，对 x² − 6x + 5 进行配方：

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

This form immediately shows that the vertex is at (3, −4) and that the minimum value is −4.

这种形式直接表明顶点为 (3, −4)，最小值为 −4。

5. Algebraic Fractions | 代数分式

To simplify an algebraic fraction, factorise the numerator and denominator first, then cancel common factors. Never cancel terms across addition or subtraction.

要化简代数分式，首先对分子和分母因式分解，然后约去公因式。切勿在加减号两侧直接约去项。

(x² − 4)/(x² + x − 6) = (x − 2)(x + 2)/((x + 3)(x − 2)) = (x + 2)/(x + 3), x ≠ 2

When adding or subtracting, use the lowest common denominator. For example, 1/(x + 1) + 2/(x − 1) = (x − 1 + 2x + 2)/((x + 1)(x − 1)) = (3x + 1)/(x² − 1).

进行加减运算时，使用最小公分母。例如，1/(x + 1) + 2/(x − 1) = (x − 1 + 2x + 2)/((x + 1)(x − 1)) = (3x + 1)/(x² − 1)。

When multiplying, multiply numerators and denominators directly. When dividing, multiply by the reciprocal of the second fraction.

乘法运算时，分子乘分子、分母乘分母。除法运算时，乘以第二个分式的倒数。

6. Polynomial Division | 多项式除法

Polynomial division is used to factorise cubics and higher-degree polynomials. Edexcel questions often ask you to divide a cubic by a linear factor.

多项式除法用于因式分解三次及更高次多项式。Edexcel 题目经常要求你将三次多项式除以一个一次因式。

Use long division just as with numbers. For example, divide x³ − 2x² − 4x + 8 by x − 2:

像数字除法一样使用长除法。例如，用 x − 2 除 x³ − 2x² − 4x + 8：

x³ − 2x² − 4x + 8 = (x − 2)(x² − 4) = (x − 2)(x − 2)(x + 2)

Here the quotient is x² − 4 and the remainder is 0. A zero remainder means x − 2 is a factor.

这里商为 x² − 4，余数为 0。余数为 0 意味着 x − 2 是一个因式。

Always write the result as dividend = divisor × quotient + remainder.

始终将结果写为：被除式 = 除式 × 商式 + 余式。

7. Factor Theorem and Remainder Theorem | 因式定理与余式定理

The remainder theorem states that when a polynomial f(x) is divided by (x − a), the remainder is f(a).

余式定理指出，当多项式 f(x) 除以 (x − a) 时，余数为 f(a)。

The factor theorem follows: if f(a) = 0, then (x − a) is a factor of f(x).

因式定理由此得出：如果 f(a) = 0，那么 (x − a) 是 f(x) 的一个因式。

For example, let f(x) = x³ − 4x² + x + 6. Since f(2) = 8 − 16 + 2 + 6 = 0, (x − 2) is a factor.

例如，设 f(x) = x³ − 4x² + x + 6。由于 f(2) = 8 − 16 + 2 + 6 = 0，所以 (x − 2) 是一个因式。

You can then divide or compare coefficients to find the remaining quadratic factor.

然后你可以使用除法或比较系数法求出剩余的二次因式。

8. Negative and Fractional Indices | 负指数与分数指数

Expressions involving roots and reciprocals can be rewritten using index notation. This makes differentiation and simplification more systematic.

涉及根式和倒数的表达式可以改写为指数形式。这使得求导和化简更加系统化。

1/x² = x⁻², √x = x^(1/2), 1/√x = x^(-1/2), ³√x = x^(1/3)

For example, simplify (x³ × √x) / x⁻²:

例如，化简 (x³ × √x) / x⁻²：

(x³ × x^(1/2)) / x⁻² = x^(3 + 1/2 + 2) = x^(11/2)

Remember that a negative index does not make the value negative; it makes it a reciprocal.

记住负指数不会使数值变为负数，而是使其成为倒数。

9. Surds and Rationalising Denominators | 根式与有理化分母

Surds are irrational roots such as √2 or √5. The key simplification rule is √(a × b) = √a × √b, provided a and b are non-negative.

根式是无理数根，如 √2 或 √5。关键的化简规则是 √(a × b) = √a × √b，其中 a 和 b 均为非负数。

For example, √72 = √(36 × 2) = 6√2. Always look for the largest square factor.

例如，√72 = √(36 × 2) = 6√2。始终寻找最大的平方因数。

To rationalise a denominator such as 1/√5, multiply numerator and denominator by √5 to get √5/5.

要对分母如 1/√5 有理化，将分子和分母同时乘以 √5，得到 √5/5。

For a denominator such as 1/(3 + √2), multiply by the conjugate 3 − √2:

对于分母如 1/(3 + √2)，乘以其共轭式 3 − √2：

1/(3 + √2) = (3 − √2)/((3 + √2)(3 − √2)) = (3 − √2)/(9 − 2) = (3 − √2)/7

Rationalising makes exact answers easier to compare and mark.

有理化使精确答案更易于比较和评分。

10. Simplifying Mixed Expressions | 化简混合表达式

Exam questions often combine several skills: factorise first, rewrite using index laws or surds, then cancel or rationalise.

考试题目经常综合考查多项技能：先因式分解，再用指数定律或根式改写，然后约分或有理化。

For example, simplify (x² − 9)/(x² + 6x + 9) × (x + 3)/(x − 3):

例如，化简 (x² − 9)/(x² + 6x + 9) × (x + 3)/(x − 3)：

(x² − 9)/(x² + 6x + 9) × (x + 3)/(x − 3) = ((x − 3)(x + 3))/(x + 3)² × (x + 3)/(x − 3) = 1

Always state any excluded values, here x ≠ 3 and x ≠ −3, because the original denominator cannot be zero.

始终注明任何排除值，此处为 x ≠ 3 且 x ≠ −3，因为原分母不能为零。

11. Common Mistakes and Exam Tips | 常见错误与应试技巧

A common error is to cancel terms incorrectly, for example writing (x + 3)/(x + 5) = 3/5. You can only cancel factors, not individual terms.

一个常见错误是错误地约去项，例如把 (x + 3)/(x + 5) 写成 3/5。你只能约去因式，不能约去单独项。

Another mistake is forgetting that (a + b)² ≠ a² + b². The correct expansion is a² + 2ab + b².

另一个错误是忘记 (a + b)² ≠ a² + b²。正确的展开式为 a² + 2ab + b²。

When rationalising, always multiply by the conjugate, and when dividing polynomials, include a zero placeholder for missing powers.

有理化时，始终乘以共轭式；进行多项式除法时，对缺失的幂次留出零占位。

In exams, show clear factorisation steps before cancelling to gain method marks even if a simplification error occurs later.

考试中，在约分前展示清晰的因式分解步骤，这样即使后续化简出现错误，也能获得方法分。

12. Summary and Practice Checklist | 总结与练习清单

Algebraic expressions underpin most of A-Level Pure Mathematics. Use the following checklist before your Edexcel exam.

代数表达式是 A-Level 纯数学大部分内容的基础。在 Edexcel 考试前，请使用以下清单进行自查。

I can apply all index laws including negative and fractional powers. ——— 我能运用所有指数定律，包括负指数和分数指数。
I can expand and factorise quadratics and cubics. ——— 我能展开并因式分解二次式和三次式。
I can complete the square and identify the vertex. ——— 我能配方并确定顶点。
I can simplify, add, subtract, multiply and divide algebraic fractions. ——— 我能化简、加减、乘除代数分式。
I can divide polynomials and use the factor theorem. ——— 我能进行多项式除法并运用因式定理。
I can manipulate surds and rationalise denominators. ——— 我能处理根式并对分母有理化。

Work through mixed exercises from past papers, because exam questions rarely isolate a single skill. Accuracy and logical setting-out are just as important as the final answer.

请完成历年真题中的混合练习，因为考试题目很少只考查单一技能。准确性、逻辑书写与最终答案同样重要。

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Algebraic Expressions | 代数表达式