An algebraic expression is a combination of constants, variables, and mathematical operations (+, -, x, /). Unlike equations, expressions do not have an equals sign — they represent a quantity.
- Key Vocabulary:
- Variable: a symbol (like x, y, a) representing an unknown or changing value
- Constant: a fixed number (like 5, -3, 0.7)
- Term: a single part of an expression separated by + or - signs (e.g. 3x, -5y2, 7)
- Coefficient: the numerical factor of a term (in 4x, the coefficient is 4)
- Like terms: terms with the same variable and the same power (e.g. 3x and -7x are like terms)
- Unlike terms: terms with different variables or different powers
- Types of Algebraic Expressions:
- Monomial: has one term (e.g. 5x, -3y2)
- Binomial: has two terms (e.g. x + 3, 2a - b)
- Trinomial: has three terms (e.g. x2 + 2x - 1)
- Polynomial: has one or more terms
Addition and Subtraction:
Add or subtract only like terms. Combine their coefficients and keep the variable part.
(3x + 4y) + (2x - y) = (3+2)x + (4-1)y = 5x + 3y.
- Multiplication:
- Monomial x monomial: multiply coefficients and add powers of like variables: 3x x 4x = 12x2.
- Monomial x polynomial: use distributive property: 2a(a + 3) = 2a2 + 6a.
Finding the Value of an Expression:
Substitute the given value of the variable and simplify.
Simplify 5x + 3y - 2x + 7y.
Combine like terms: (5-2)x + (3+7)y = 3x + 10y.
Subtract (3a - 4b) from (7a + 2b).
= (7a + 2b) - (3a - 4b) = 7a + 2b - 3a + 4b = 4a + 6b.
Find the value of 2x2 - 3x + 5 when x = 2.
= 2(4) - 3(2) + 5 = 8 - 6 + 5 = 7.
Multiply 3x by (2x - 5).
= 3x x 2x + 3x x (-5) = 6x2 - 15x.
Simplify (4p + 3q) + (2p - q) - (p + 4q).
= 4p + 3q + 2p - q - p - 4q = 5p - 2q.
Write an expression for "5 more than twice x".
= 2x + 5.
If a = 3 and b = -2, find the value of 3a2 - 2ab + b2.
= 3(9) - 2(3)(-2) + (-2)2 = 27 + 12 + 4 = 43.
Summary
- Like terms have identical variable parts.
- Only like terms can be added or subtracted.
- When subtracting an expression, distribute the negative sign to all its terms.
Common mistakes
When subtracting a bracket, remember to change the sign of EVERY term inside the bracket. For example, -(3a - 4b) = -3a + 4b (not -3a - 4b). Also, do not add unlike terms — 3x + 4y cannot be simplified.
Summary
Algebraic expressions use variables and operations to generalise mathematical relationships. Key skills are identifying like terms, adding/subtracting expressions, multiplying monomials, and evaluating expressions by substitution.