Building Blocks of Algebra: Algebraic Expressions
Algebraic expressions use variables, constants, and operations to represent mathematical relationships in a general form. Mastering how to write, simplify, factor, and evaluate these expressions is the foundation for solving equations, graphing functions, and modeling real-world situations throughout algebra and beyond.
Components of Algebraic Expressions
This section covers the core skills for working with algebraic expressions:
- Terms, Coefficients & Constants: A term is a product of numbers and variables (like 3x²); the coefficient is the numerical part (3); a constant has no variable (like 7).
- Combining Like Terms: Adding or subtracting terms that share the same variable and exponent to simplify an expression.
- Distributive Property: Multiplying a factor across a sum or difference inside parentheses: a(b + c) = ab + ac.
- Factoring Common Factors: Reversing distribution by pulling out the greatest common factor from all terms.
Examples of Algebraic Expressions
Terms, Coefficients & Constants Examples
- In the expression 5x² - 3x + 7, the terms are 5x², -3x, and 7. The coefficients are 5 and -3, and 7 is the constant.
- In 4ab + 2a - 9, the coefficient of ab is 4, the coefficient of a is 2, and -9 is the constant term.
- Identify the parts of -2y³ + y - 1: the coefficient of y³ is -2, the coefficient of y is 1 (implied), and -1 is the constant.
Combining Like Terms Examples
- Simplify 4x + 7 - 2x + 3: Group like terms to get (4x - 2x) + (7 + 3) = 2x + 10.
- Simplify 3a² + 5a - a² + 2a: Group to get (3a² - a²) + (5a + 2a) = 2a² + 7a.
- Simplify 6y - 4 + 2y + 9 - y: Group to get (6y + 2y - y) + (-4 + 9) = 7y + 5.
Distributive Property Examples
- Expand 3(x + 4): Multiply to get 3x + 12.
- Expand -2(5a - 3): Multiply to get -10a + 6.
- Expand 4(2x² - x + 5): Multiply each term to get 8x² - 4x + 20.
Factoring Common Factors Examples
- Factor 6x + 18: The GCF is 6, so factor to get 6(x + 3).
- Factor 10a² - 15a: The GCF is 5a, so factor to get 5a(2a - 3).
- Factor 12y³ + 8y² - 4y: The GCF is 4y, so factor to get 4y(3y² + 2y - 1).