Understanding Algebraic Expressions
Algebraic expressions are mathematical phrases that contain numbers, variables, and operations. They form the foundation of algebra and are essential for solving equations and modeling real-world problems.
Components of Algebraic Expressions
Terms
A term is a single mathematical expression that may be a number, variable, or combination of numbers and variables multiplied together.
- Constant terms: Numbers without variables (e.g., 5, -3)
- Variable terms: Variables with or without coefficients (e.g., x, 3y, -2z)
- Like terms: Terms with identical variables and exponents
Coefficients
The numerical factor of a term containing variables.
- In 3x², the coefficient is 3
- In -5y, the coefficient is -5
- In x, the coefficient is 1 (understood)
Variables
Letters that represent unknown or changeable values.
- Common variables: x, y, z, a, b, c
- Can represent any real number
- Allow for generalization of mathematical relationships
Types of Algebraic Expressions
Polynomial Expressions
Expressions with terms that have non-negative integer exponents.
- Monomial: One term (e.g., 3x²)
- Binomial: Two terms (e.g., x + 5)
- Trinomial: Three terms (e.g., x² + 2x + 1)
- Polynomial: Multiple terms (e.g., x³ + 2x² - x + 3)
Rational Expressions
Fractions where numerator and denominator are polynomials.
- Example: (x² + 1)/(x - 2)
- Domain restrictions where denominator equals zero
- Can be simplified by factoring
Radical Expressions
Expressions containing square roots, cube roots, or other radicals.
- Example: √(x² + 4)
- Can often be simplified
- Domain restrictions for even roots
Operations with Algebraic Expressions
Addition and Subtraction
Combine like terms by adding or subtracting their coefficients.
- 3x + 5x = 8x
- 7y² - 2y² = 5y²
- Unlike terms cannot be combined: 3x + 5y remains as is
Multiplication
Use distributive property and combine like terms.
- 3x(2x + 1) = 6x² + 3x
- (x + 2)(x + 3) = x² + 5x + 6
- FOIL method for binomials
Division
Factor when possible and cancel common factors.
- Polynomial long division for complex cases
- Factor and simplify rational expressions
- Check for domain restrictions
Simplification Techniques
Combining Like Terms
- Identify terms with identical variables and exponents
- Add or subtract their coefficients
- Keep the variable part unchanged
Factoring
- Common factors: 6x² + 9x = 3x(2x + 3)
- Difference of squares: x² - 4 = (x + 2)(x - 2)
- Perfect square trinomials: x² + 6x + 9 = (x + 3)²
- Quadratic factoring: x² + 5x + 6 = (x + 2)(x + 3)
Expanding
- Distribute multiplication over addition/subtraction
- Use special products (FOIL, perfect squares)
- Combine like terms after expansion
Applications
Problem Solving
- Modeling real-world situations
- Setting up equations from word problems
- Representing relationships between quantities
Function Analysis
- Defining mathematical functions
- Finding domain and range
- Analyzing behavior and properties
Equation Solving
- Setting expressions equal to solve equations
- Finding zeros and roots
- Solving systems of equations
