Algebraic Expressions

Definition

Algebraic expressions are mathematical phrases that combine variables, constants, and mathematical operations. They can be simplified, evaluated, and manipulated using algebraic rules.

Types of Algebraic Expressions

• Monomials: single-term expressions, e.g., 2x^2, 5y
• Binomials: two-term expressions, e.g., x + 3, 2x - 4
• Polynomials: expressions with multiple terms, e.g., x^2 + 3x - 2, 2x^3 - 5x^2 + x

Properties of Algebraic Expressions

• Commutative Property: the order of variables and constants does not change the result, e.g., 2x + 3 = 3 + 2x
• Associative Property: the order of operations does not change the result, e.g., (2x + 3) + 2 = 2x + (3 + 2)
• Distributive Property: a single value can be distributed to multiple terms, e.g., 2(x + 3) = 2x + 6

Simplifying Algebraic Expressions

• Combining Like Terms: combine terms with the same variable and coefficient, e.g., 2x + 3x = 5x
• Removing Parentheses: apply the distributive property to remove parentheses, e.g., 2(x + 3) = 2x + 6

Evaluating Algebraic Expressions

• Substitute values for variables to evaluate the expression, e.g., if x = 2, then 2x + 3 = 2(2) + 3 = 7

代数表达式定义

• 代数表达式是将变量、常数和数学运算符结合起来的数学短语
• 可以根据代数规则对其进行简化、求值和操作

代数表达式类型

• 单项式: 单个项的表达式，例如 2x^2、5y
• 二项式: 两个项的表达式，例如 x + 3、2x - 4
• 多项式: 多个项的表达式，例如 x^2 + 3x - 2、2x^3 - 5x^2 + x

代数表达式性质

• 交换性: 变量和常数的顺序不改变结果，例如 2x + 3 = 3 + 2x
• 结合性: 运算顺序不改变结果，例如 (2x + 3) + 2 = 2x + (3 + 2)
• 分布性: 单个值可以分布到多个项，例如 2(x + 3) = 2x + 6

简化代数表达式

• 合并同类项: 将具有相同变量和系数的项合并，例如 2x + 3x = 5x
• 去括号: 应用分布性来去除括号，例如 2(x + 3) = 2x + 6

评估代数表达式

• 将变量值代入表达式以评估结果，例如如果 x = 2，则 2x + 3 = 2(2) + 3 = 7