1.1 变量、系数与常数项 Variables, Coefficients, and Constant Terms
代数式（algebraic expression）是由数字、字母（变量）和运算符号（+、−、×、÷）组成的数学表达式。
An algebraic expression is a mathematical phrase that can contain numbers, variables (letters), and operations (+, −, ×, ÷).
系数（Coefficient）：变量前面的数字因数，如 5x 中的 5
常数项（Constant）：不含变量的固定数，如 3x + 7 中的 7
项（Term）：用加号或减号隔开的各部分，如 3x + 2y − 5 有三项
Coefficient: the numerical factor of a variable (e.g., 5 in 5x)
Constant: a fixed number with no variable (e.g., 7 in 3x + 7)
Term: parts separated by + or − signs (e.g., 3x + 2y − 5 has three terms)
举例：在代数式 4x² − 3xy + 7y − 5 中：
- 4x² 的系数是 4，变量是 x The coefficient of 4x² is 4
- −3xy 的系数是 −3（注意符号！） The coefficient of −3xy is −3 (include the sign!)
- 7y 的系数是 7 The coefficient of 7y is 7
- −5 是常数项 −5 is the constant term
The coefficient always includes the sign! The coefficient of −3x is −3, not 3. This is a common trap on the AMC.
1.2 同类项 Like Terms
同类项（like terms）是指含有相同变量且每个变量的指数都相同的项。只有同类项才能合并。
Like terms are terms that have the same variables raised to the same powers. Only like terms can be combined.
| 是否为同类项？ | Are they like terms? | 说明 |
|---|---|---|
| 3x 和 5x | ✅ 是 | 变量相同，指数相同 |
| 2x² 和 7x² | ✅ 是 | 变量的幂次相同 |
| 4xy 和 9xy | ✅ 是 | 变量组合和指数都相同 |
| 3x 和 3x² | ❌ 否 | 变量指数不同 |
| 2x 和 2y | ❌ 否 | 变量不同 |
| 5xy 和 5yx | ✅ 是 | xy 和 yx 是相同的（乘法交换律） |
Different coefficients don't prevent terms from being like terms! 3x and 7x are like terms and combine to 10x.
2.1 合并同类项 Combining Like Terms
合并同类项的法则是：将同类项的系数相加减，变量部分保持不变。
To combine like terms: add or subtract the coefficients while keeping the variable part the same.
示例：
- 3x + 5x = (3+5)x = 8x
- 7a − 2a = (7−2)a = 5a
- 4x² + 3x − x² + 2x = (4−1)x² + (3+2)x = 3x² + 5x
- 6xy − 2xy + 5 = 4xy + 5（5 没有同类项，保留）
Before combining, mark like terms (underline, circle, etc.) to avoid missing or mixing terms. This is key to reducing errors!
2.2 去括号规则 Rules for Removing Parentheses
当代数式中出现括号时，需要根据括号前的符号决定如何去掉括号：
When parentheses appear in expressions, the sign before them determines how to remove them:
−(a + b − c) = −a − b + c (负号变号)
示例：
- 2x + (3y − 4) = 2x + 3y − 4
- 2x − (3y − 4) = 2x − 3y + 4
- 3(a + 2b) − (a − 5b) = 3a + 6b − a + 5b = 2a + 11b
Common mistake: With a negative sign, ALL signs inside must flip. −(3−5) = −3+5 = 2, NOT −3−5 = −8.
3.1 分配律 The Distributive Property
分配律（distributive property）是代数式乘法的基础法则：
The distributive property is the fundamental rule for multiplying expressions:
a(b − c) = ab − ac
示例：
- 3(2x + 5) = 3·2x + 3·5 = 6x + 15
- −2(4x − 3) = −2·4x + (−2)·(−3) = −8x + 6
- 5(2a + 3) − 2(a − 1) = 10a + 15 − 2a + 2 = 8a + 17
When the number outside is negative, pay extra attention to signs! −2 × (−3) = +6, not −6.
3.2 指数法则初步 Basic Exponent Rules
AMC 8 中需要掌握最基本的指数运算法则：
You need to master the most basic exponent rules for the AMC 8:
| 法则 | Rule | 示例 | Example |
|---|---|---|---|
| 同底数相乘 | xa · xb = xa+b | x³ · x² = x⁵ | Add exponents |
| 同底数相除 | xa ÷ xb = xa−b | x⁵ ÷ x² = x³ | Subtract exponents |
| 幂的乘方 | (xa)b = xab | (x³)² = x⁶ | Multiply exponents |
| 负指数 | x−n = 1/xn | x−2 = 1/x² | Reciprocal |
x¹ = x
1n = 1
Exponent rules only apply to same bases! x² · y³ ≠ (xy)⁵. Different bases cannot be combined this way.
化简：5x − 3 + 2x + 7 − xSimplify: 5x − 3 + 2x + 7 − x
合并常数项：(−3 + 7) = 4
结果：6x + 4 Combine x terms: (5+2−1)x = 6x. Combine constants: −3+7 = 4. Result: 6x + 4.
化简：3(2a − 1) − 2(a + 4)Simplify: 3(2a − 1) − 2(a + 4)
展开第二个括号（注意负号）：−2(a + 4) = −2a − 8
合并：(6a − 2a) + (−3 − 8) = 4a − 11 Expand: 3(2a−1) = 6a−3. Note the minus sign: −2(a+4) = −2a−8. Combine: (6a−2a)+(−3−8) = 4a−11.
化简：x³ · x⁴ − 2x⁵ + x² · x⁵Simplify: x³ · x⁴ − 2x⁵ + x² · x⁵
x² · x⁵ = x2+5 = x⁷
代入：x⁷ − 2x⁵ + x⁷ = 2x⁷ − 2x⁵
也可以提取公因式：2x⁵(x² − 1) x³·x⁴ = x⁷, x²·x⁵ = x⁷. So: x⁷ − 2x⁵ + x⁷ = 2x⁷ − 2x⁵ = 2x⁵(x² − 1).
若 3x + 7 = 22，则 5x − 3 的值是多少？If 3x + 7 = 22, what is the value of 5x − 3?
代入 5x − 3 = 5(5) − 3 = 25 − 3 = 22 Solve: 3x + 7 = 22 → 3x = 15 → x = 5. Substitute: 5(5) − 3 = 25 − 3 = 22.
化简：2(3x − 4) − 3(x + 2) + 5(x − 1)Simplify: 2(3x − 4) − 3(x + 2) + 5(x − 1)
2(3x − 4) = 6x − 8
−3(x + 2) = −3x − 6
5(x − 1) = 5x − 5
合并同类项：(6x − 3x + 5x) + (−8 − 6 − 5) = 8x − 19 Expand each: 2(3x−4)=6x−8, −3(x+2)=−3x−6, 5(x−1)=5x−5. Combine: (6−3+5)x + (−8−6−5) = 8x − 19.
第1题 化简：7a + 3b − 4a + 2bSimplify: 7a + 3b − 4a + 2b
第2题 化简：−2(3x − 5) + 4(x + 1)Simplify: −2(3x − 5) + 4(x + 1)
第3题 若 x = 3，则 2x² − 3x + 1 的值是多少？If x = 3, what is the value of 2x² − 3x + 1?
第4题 化简：(2x)³ ÷ (2x) 的结果是多少？What is (2x)³ ÷ (2x) simplified?
第5题 化简：3x(x − 2) − 2x(x + 1)，结果中 x² 的系数是多少？Simplify 3x(x−2) − 2x(x+1). What is the coefficient of x²?
第6题 若 2(x + 3) − 4(x − 1) = 10，则 x 的值是多少？If 2(x+3) − 4(x−1) = 10, what is x?
第7题 化简：4a²b · 3ab² ÷ (6ab)Simplify: 4a²b · 3ab² ÷ (6ab)
第8题 若 a + b = 5 且 a − b = 3，则 a² − b² 的值是多少？If a+b=5 and a−b=3, what is a²−b²?