Expanding Brackets | 展开括号

📚 Expanding Brackets | 展开括号

Expanding brackets is one of the most fundamental algebraic skills in the Edexcel A Level Mathematics course. It means removing grouping symbols such as ( ) and [ ] by multiplying each term inside the bracket by the factor outside, then simplifying by collecting like terms. This skill appears in quadratics, functions, calculus, and binomial expansions.

展开括号是 Edexcel A Level 数学课程中最基本的代数技能之一。它是指通过将括号外的因式与括号内的每一项相乘，去掉 ( )、[ ] 等分组符号，然后合并同类项进行化简。这一技能在二次函数、函数、微积分和二项式展开中都会出现。

1. The Meaning of Expanding Brackets | 展开括号的含义

At its core, expanding brackets applies the distributive law: a(b + c) = ab + ac. Brackets show that all terms inside are grouped together and must be multiplied by the same factor outside.

展开括号的核心是应用分配律：a(b + c) = ab + ac。括号表示里面的所有项是一个整体，必须同乘括号外的同一个因式。

a(b + c) = ab + ac

When signs and coefficients are involved, the same principle holds, but negative factors change the signs of the terms inside. This is why expanding with negative numbers needs extra care.

当涉及符号和系数时，原理保持不变，但负的因式会改变括号内各项的符号。这就是为什么含有负数的展开需要格外小心。

2. Expanding a Single Bracket | 展开单项式乘括号

To expand a single bracket, multiply the term outside by every term inside the bracket. For example, 3x(2x − 5) gives 6x² − 15x because 3x × 2x = 6x² and 3x × (−5) = −15x.

展开单项式乘括号时，用括号外的项乘以括号内的每一项。例如，3x(2x − 5) 得到 6x² − 15x，因为 3x × 2x = 6x²，而 3x × (−5) = −15x。

3x(2x − 5) = 3x × 2x − 3x × 5 = 6x² − 15x

Always multiply coefficients and add exponents when multiplying powers of x. For instance, x × x = x², and 4y × 3y² = 12y³. Do not forget that the exponent applies only to the variable.

同底数幂相乘时，系数相乘、指数相加。例如，x × x = x²，4y × 3y² = 12y³。不要忘记指数只作用于变量。

A more complex example is 4y(3y² − 2y + 7), which expands to 12y³ − 8y² + 28y. Each term inside the bracket is multiplied by 4y in turn.

一个更复杂的例子是 4y(3y² − 2y + 7)，展开后得到 12y³ − 8y² + 28y。括号内的每一项依次乘以 4y。

3. Expanding Double Brackets | 展开两个二项式

To expand two binomials such as (x + 3)(x + 2), every term in the first bracket must multiply every term in the second bracket. This is often remembered by FOIL: First, Outer, Inner, Last.

展开两个二项式（如 (x + 3)(x + 2)）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。通常用 FOIL 记忆：首项、外项、内项、末项。

(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6

The grid method is a visual alternative. Write one bracket along the top and one along the side, then fill in the products and add them together. This helps reduce missed terms when the signs are tricky.

网格法是一种直观的替代方法。将一个括号写在顶部，另一个写在侧边，然后填写乘积并相加。当符号容易混淆时，这有助于减少漏项。

×	x	+2
x	x²	+2x
+3	+3x	+6

From the grid, the four terms are x², 2x, 3x and 6. Adding the two x terms gives the simplified quadratic x² + 5x + 6.

从网格中得到四项：x²、2x、3x 和 6。将两个 x 项相加，得到化简后的二次式 x² + 5x + 6。

4. Expanding with Negative Signs | 含有负号的展开

Negative signs are a common source of error. When expanding −2x(x − 4), the factor −2x must multiply both x and −4, producing −2x² + 8x. Note how the product of two negatives gives a positive term.

负号是常见的错误来源。展开 −2x(x − 4) 时，因式 −2x 必须同时乘以 x 和 −4，得到 −2x² + 8x。注意两个负数相乘得到正项。

−2x(x − 4) = −2x² + 8x

When a bracket is preceded by a minus sign, rewrite it as multiplication by −1 before expanding, or mentally flip the sign of every term inside. For example, −(2x − 5) = −2x + 5.

当括号前是负号时，可以先将其改写为乘以 −1 再展开，或在心中翻转括号内每一项的符号。例如，−(2x − 5) = −2x + 5。

In double brackets, handle signs systematically: (x − 4)(x + 3) gives x² + 3x − 4x − 12, which simplifies to x² − x − 12. The inner and outer products have opposite signs, so only their coefficients are combined.

在双括号中，要有条理地处理符号：(x − 4)(x + 3) 得到 x² + 3x − 4x − 12，化简为 x² − x − 12。内项和外项符号相反，因此只需合并它们的系数。

5. Expanding Polynomial Expressions | 展开多项式表达式

The same method extends to brackets with more than two terms. For example, (x + 2)(x² − 3x + 1) requires multiplying x by all three terms in the second bracket, then 2 by all three terms.

同样的方法可以推广到含有两项以上的括号。例如 (x + 2)(x² − 3x + 1) 需要将 x 乘以第二个括号中的三项，再将 2 乘以这三项。

(x + 2)(x² − 3x + 1) = x³ − 3x² + x + 2x² − 6x + 2 = x³ − x² − 5x + 2

Collecting like terms after multiplication is essential: terms with x³, x², x and constants must be grouped separately. Work from the highest power to the lowest to keep the final expression tidy.

乘法后合并同类项至关重要：x³ 项、x² 项、x 项和常数项必须分别合并。按照从高次幂到低次幂的顺序整理，可以使最终表达式保持整洁。

Another useful example is (2x − 1)(x² + 4x − 3), which expands to 2x³ + 8x² − 6x − x² − 4x + 3, and then simplifies to 2x³ + 7x² − 10x + 3.

另一个有用的例子是 (2x − 1)(x² + 4x − 3)，它展开为 2x³ + 8x² − 6x − x² − 4x + 3，然后化简为 2x³ + 7x² − 10x + 3。

6. Special Product: Squaring a Binomial | 特殊乘积：二项式平方

A squared binomial expands using the formula (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². Do not simply square each term inside; the cross term 2ab is required.

二项式平方可展开为 (a + b)² = a² + 2ab + b² 以及 (a − b)² = a² − 2ab + b²。不要只对括号内每一项平方，中间的交叉项 2ab 必不可少。

(x + 5)² = x² + 10x + 25

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Expanding Brackets | 展开括号