📚 Expanding Brackets: Techniques and Methods | 整式运算：括号展开的技巧与方法
Expanding brackets is one of the most fundamental skills in algebra. It allows us to simplify expressions, solve equations, and manipulate polynomials with confidence. This guide covers the essential techniques and methods for expanding brackets clearly and accurately.
括号展开是代数中最基础的技能之一。它帮助我们简化表达式、解方程，并自信地处理多项式。本指南将系统讲解括号展开的核心技巧与方法，确保准确、高效。
1. The Distributive Law | 乘法分配律
The distributive law states that for any numbers or expressions a, b, and c:
a(b + c) = ab + ac. Every term inside the bracket must be multiplied by the term outside.
乘法分配律指出：对于任何数或表达式 a、b、c，有 a(b + c) = ab + ac。括号内的每一项都必须乘以外面的项。
a(b + c) = ab + ac
For example, 3(x + 4) = 3x + 12. The number 3 multiplies both x and 4 separately.
例如，3(x + 4) = 3x + 12。数字 3 分别与 x 和 4 相乘。
2. Monomial × Polynomial | 单项式乘多项式
When a monomial (a single term) multiplies a polynomial, apply the distributive law to each term inside the bracket. Pay attention to signs and exponents.
当单项式（一项）乘以多项式时，将分配律应用到括号内的每一项。注意符号和指数。
-2x²(3x − 5y) = -6x³ + 10x²y
Here, -2x² multiplies 3x to give -6x³, and multiplies -5y to give +10x²y.
这里，-2x² 乘以 3x 得 -6x³，乘以 -5y 得 +10x²y。
- Multiply coefficients and add exponents of like bases.
- 系数相乘，同底数指数相加。
- Keep the sign of each term: negative × negative = positive.
- 注意符号：负负得正。
3. Binomial × Binomial | 二项式乘二项式
To expand (a + b)(c + d), multiply each term in the first bracket by each term in the second bracket. A common shortcut is FOIL: First, Outer, Inner, Last.
若要展开 (a + b)(c + d)，将第一个括号中的每一项与第二个括号中的每一项相乘。常用口诀是 FOIL：首、外、内、尾。
(a + b)(c + d) = ac + ad + bc + bd
For example, (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6.
例如，(x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6。
| First | x · x = x² |
| Outer | x · (−2) = −2x |
| Inner | 3 · x = 3x |
| Last | 3 · (−2) = −6 |
4. Perfect Square Expansion | 完全平方展开
A perfect square is the square of a binomial. Memorise these formulas to expand quickly without going through FOIL step by step.
完全平方是二项式的平方。记住这些公式，可以快速展开，无需逐步 FOIL。
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
Notice that the middle term is twice the product of the two terms. For (2x + 5)², we get (2x)² + 2(2x)(5) + 5² = 4x² + 20x + 25.
注意中间项是两项乘积的两倍。对于 (2x + 5)²，得到 (2x)² + 2(2x)(5) + 5² = 4x² + 20x + 25。
5. Difference of Two Squares | 平方差公式
When two binomials have the same terms but opposite signs between them, the cross terms cancel, leaving only a difference of squares.
当两个二项式具有相同的项，但中间符号相反时，交叉项相互抵消，只剩平方差。
(a + b)(a − b) = a² − b²
For example, (3x + 2)(3x − 2) = (3x)² − 2² = 9x² − 4.
例如，(3x + 2)(3x − 2) = (3x)² − 2² = 9x² − 4。
- Check that both brackets have identical first and second terms.
- 确认两个括号的首项和末项完全相同。
- The result has only two terms, no middle term.
- 结果只有两项，没有中间项。
6. Cube Formulas | 立方公式
For 3-dimensional expansion, the formulas for (a ± b)³ are worth memorising. They appear frequently in factorisation and calculus.
对于立方展开，(a ± b)³ 的公式值得记忆。它们在因式分解和微积分中经常出现。
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
An easy pattern: the coefficients are 1, 3, 3, 1. The powers of a decrease, while powers of b increase.
记忆规律：系数为 1、3、3、1。a 的幂递减，b 的幂递增。
7. Nested Brackets | 嵌套括号
When brackets contain other brackets, expand from the inside outward. Work step by step to avoid sign and term mistakes.
当括号中还有括号时，由内向外展开。逐步运算，避免符号和项的错误。
Example: 2[x + 3(x − 1)] = 2[x + 3x − 3] = 2[4x − 3] = 8x − 6.
例如：2[x + 3(x − 1)] = 2[x + 3x − 3] = 2[4x − 3] = 8x − 6。
- First remove the inner bracket by distributing.
- 先通过分配律去掉内层括号。
- Simplify inside the outer bracket before expanding again.
- 在再次展开之前，先化简外层括号内的内容。
8. Handling Negative Signs | 处理负号
A minus sign outside a bracket changes the sign of every term inside. This is a common source of errors.
括号外的负号会改变括号内每一项的符号。这是常见错误来源。
−(a − b + c) = −a + b − c
If the bracket is multiplied by a negative monomial, distribute the monomial’s sign along with its coefficient.
若括号乘以一个负数单项式，要连同系数的符号一起分配。
9. Combining Like Terms | 合并同类项
After expanding, collect like terms: terms that have the same variable(s) raised to the same powers. This gives the final simplified form.
展开后，合并同类项：即变量及其指数完全相同的项。这得到最终最简形式。
Example: (x + 2)(x + 3) − 2x = x² + 3x + 2x + 6 − 2x = x² + 3x + 6.
例如：(x + 2)(x + 3) − 2x = x² + 3x + 2x + 6 − 2x = x² + 3x + 6。
Notice 3x + 2x − 2x simplifies to just 3x, because +2x and −2x cancel.
注意 3x + 2x − 2x 化简后只剩 3x，因为 +2x 和 −2x 相消。
10. Common Mistakes | 常见错误
Being aware of typical mistakes helps you avoid them. Here are the most frequent ones:
意识到典型错误有助于避免它们。以下是最常见的几点：
| Mistake | Correct |
| (a + b)² = a² + b² | a² + 2ab + b² |
| −(x − y) = −x − y | −x + y |
| x(x + 1) = x² + 1 | x² + x |
Always double-check that every term is multiplied, and that signs are correct.
始终检查每一项是否都相乘，符号是否正确。
11. Strategic Techniques | 策略性技巧
For longer polynomials, use the “grid method”: write each term of one polynomial along the top and each term of the other along the side of a table. Fill in each product, then sum all entries.
对于较长的多项式，可使用“表格法”：将一个多项式的各项写在表格上方，另一个写在左侧。填入每个乘积，然后求和所有格子。
| x | +2 | |
| x | x² | 2x |
| −3 | −3x | −6 |
Then sum: x² + 2x − 3x − 6 = x² − x − 6.
然后相加：x² + 2x − 3x − 6 = x² − x − 6。
12. Applications | 实际应用
Expanding brackets is not just a mechanical exercise. It is used to solve quadratic equations, interpret graphs, simplify rational expressions, and model real-world situations.
括号展开不只是机械练习。它用于解二次方程、解读图像、化简有理表达式，以及建立现实模型。
For instance, to find the area of a rectangle with length (x + 4) and width (x − 1), expand (x + 4)(x − 1) = x² + 3x − 4.
例如，求长为 (x + 4)、宽为 (x − 1) 的矩形面积，展开 (x + 4)(x − 1) = x² + 3x − 4。
Master these techniques, and you will handle algebra with ease.
掌握这些技巧，你将能轻松应对代数问题。
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