What Is an Algebraic Expression?
An algebraic expression is a mathematical phrase that combines variables (like x, y), constants (like 3, -7), and operations (addition, subtraction, multiplication, division). Unlike an equation, an expression does not have an equals sign.
Example: 3x² + 5x - 7 is an expression. 3x² + 5x - 7 = 0 is an equation.
Parts of an Expression
| Part | What It Is | Example |
|---|---|---|
| Variable | A symbol for an unknown number | x, y, z |
| Coefficient | The number multiplied by a variable | In 5x, the coefficient is 5 |
| Constant | A number with no variable | -7, 3, 12 |
| Term | Each part separated by + or - | 3x², 5x, -7 |
Types of Algebraic Expressions
Expressions are named by how many terms they have:
- Monomial - one term: 5x, -3y², 8
- Binomial - two terms: x + 3, 2a - 5b
- Trinomial - three terms: x² + 2x + 1
- Polynomial - any number of terms (general term): 4x³ - 2x² + x - 6
Operations on Algebraic Expressions
1. Adding and Subtracting
Only combine like terms - terms with the same variable and same exponent.
- 3x + 5x = 8x (same variable x)
- 4y² - y² = 3y² (same variable and exponent)
- 3x + 5y = 3x + 5y (cannot combine - different variables)
2. Multiplying
Use the distributive property: a(b + c) = ab + ac
- 2(x + 3) = 2x + 6
- 3(2x - 4) = 6x - 12
- (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 (FOIL method)
3. Dividing
Divide each term in the numerator by the denominator, or factor and cancel common factors.
- (6x + 9) / 3 = 2x + 3
- (x² - 4) / (x + 2) = x - 2
Step 1 - Distribute: 6x - 2 + 4x + 20
Step 2 - Combine like terms: (6x + 4x) + (-2 + 20) = 10x + 18
FOIL Method for Multiplying Binomials
When multiplying two binomials, use FOIL: First, Outer, Inner, Last.
Example: (x + 4)(x - 2)
- First: x × x = x²
- Outer: x × (-2) = -2x
- Inner: 4 × x = 4x
- Last: 4 × (-2) = -8
Combine: x² - 2x + 4x - 8 = x² + 2x - 8
Common Mistakes
- Combining unlike terms: 3x + 2y does NOT equal 5xy. You can only combine like terms.
- Forgetting to distribute: 2(x + 3) is NOT 2x + 3. It's 2x + 6.
- Sign errors: When subtracting, distribute the negative: -(x - 4) = -x + 4, not -x - 4.
Try our Solve for X Calculator or Polynomial Calculator for more practice.