Expanding Brackets and Simplifying Expressions | 去括号与化简表达式

📚 Expanding Brackets and Simplifying Expressions | 去括号与化简表达式

Expanding brackets is a fundamental skill in KS3 algebra that allows you to rewrite expressions without parentheses. Mastering this technique is crucial for solving equations, simplifying expressions, and understanding more advanced topics like factorising and quadratic functions. This article will guide you through the process, from simple single brackets to more complex double brackets, with clear examples and practice problems.

去括号是 KS3 代数中的基础技能，能够让你将带括号的表达式改写成没有括号的形式。掌握这项技术对于解方程、化简表达式以及理解更高级的主题（如因式分解和二次函数）至关重要。本文将通过清晰的示例和练习题，引导你完成从简单的单括号到更复杂的双括号的学习过程。

1. What Are Brackets in Algebra? | 什么是代数中的括号？

In algebra, brackets (parentheses) are used to indicate that the terms inside should be treated as a group. For example, in the expression 3(x + 2), the brackets tell us that the entire quantity ‘x + 2’ is multiplied by 3. Without brackets, 3x + 2 would mean only x is multiplied by 3. Brackets help define the order of operations, ensuring that calculations inside them are performed first, following the BIDMAS/BODMAS rule.

在代数中，括号（小括号）用于表示括号内的各项应被视为一个整体。例如，在表达式 3(x + 2) 中，括号告诉我们整个 ‘x + 2’ 都要乘以 3。如果没有括号，3x + 2 就只表示 x 乘以 3。括号有助于明确运算顺序，确保先计算括号内的内容，遵循 BIDMAS/BODMAS 法则。

2. The Distributive Property | 分配律

The key rule for expanding brackets is the distributive property, which states that multiplying a sum by a number is the same as multiplying each addend individually and then adding the results. Algebraically: a(b + c) = ab + ac. This means you multiply the term outside the bracket by every term inside the bracket. For example, 5(y + 3) = 5 × y + 5 × 3 = 5y + 15.

展开括号的关键法则是分配律，即一个数与一个和相乘，等于这个数分别乘以每个加数，然后将结果相加。代数表示为：a(b + c) = ab + ac。这意味着你要将括号外的项乘以括号内的每一项。例如，5(y + 3) = 5 × y + 5 × 3 = 5y + 15。

The distributive property also works for subtraction: a(b – c) = ab – ac. Always remember to multiply every term inside the bracket, even if it is a constant or a variable with a coefficient.

分配律也适用于减法：a(b – c) = ab – ac。始终记住要乘以括号内的每一项，无论它是常数还是带有系数的变量。

3. Expanding a Single Bracket | 展开单个括号

To expand an expression like 4(2x + 5), multiply the 4 by everything inside: 4 × 2x = 8x, and 4 × 5 = 20. So, 4(2x + 5) = 8x + 20. Similarly, 3(4m – 7) = 3 × 4m – 3 × 7 = 12m – 21. The process is straightforward, but careful arithmetic is essential, especially when dealing with fractions or decimals. For instance, ½(8x + 6) = 4x + 3 (since ½ × 8x = 4x, ½ × 6 = 3).

要展开如 4(2x + 5) 这样的表达式，用 4 乘以括号内的每一项：4 × 2x = 8x，4 × 5 = 20。因此，4(2x + 5) = 8x + 20。类似地，3(4m – 7) = 3 × 4m – 3 × 7 = 12m – 21。过程很简单，但仔细的算术运算是必要的，尤其是在处理分数或小数时。例如，½(8x + 6) = 4x + 3（因为 ½ × 8x = 4x，½ × 6 = 3）。

If the bracket contains more than two terms, the same rule applies: multiply the outside term by each term inside. For example, 2(a + 3b – c) = 2a + 6b – 2c.

如果括号内有超过两项，同样的规则适用：用外部的项乘以内部的每一项。例如，2(a + 3b – c) = 2a + 6b – 2c。

4. Expanding with Negative Numbers | 负数的展开

When the term outside the bracket is negative, you must be careful with signs. Remember that multiplying a negative by a positive gives a negative, and a negative by a negative gives a positive. For example, –3(x + 4) = –3x – 12. And –2(5 – y) = –2 × 5 + (–2) × (–y) = –10 + 2y. A common error is to forget to change the sign of the last term. Always distribute the negative sign to every term inside.

当括号外的项是负数时，你必须注意符号。记住，负数乘以正数得负数，负数乘以负数得正数。例如，–3(x + 4) = –3x – 12。而 –2(5 – y) = –2 × 5 + (–2) × (–y) = –10 + 2y。一个常见错误是忘记改变最后一项的符号。永远要把负号分配给括号内的每一项。

Consider –(2x – 3). This is the same as –1 × (2x – 3) = –2x + 3. Understanding this helps avoid mistakes when expanding expressions with a leading minus sign.

考虑 –(2x – 3)。这等同于 –1 × (2x – 3) = –2x + 3。理解这一点有助于在展开带有前置减号的表达式时避免错误。

5. Expanding and Simplifying | 展开后化简

Often, after expanding brackets, you will need to collect like terms to simplify the expression. Like terms are those that have the same variable raised to the same power. For example, in 3(x + 2) +

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