In this chapter, you will learn
- —Understand variables, constants, and algebraic expressions
- —Identify and classify algebraic expressions (monomials, binomials, polynomials)
- —Perform addition and subtraction of algebraic expressions
- —Multiply and simplify algebraic expressions
- —Apply algebraic expressions in solving word problems
Variables, Constants, and Algebraic Expressions
An algebraic expression is a mathematical phrase containing variables, constants, and operations.
Definitions:
- Variable: A letter (x, y, a, b) representing an unknown number
- Constant: A fixed number (2, 5, -3)
- Algebraic Expression: A combination like 2x + 3, x² - 4, 5a + 2b - 7
Examples:
- 2x (monomial: one term)
- 3x + 5 (binomial: two terms)
- x² + 2x + 1 (trinomial: three terms)
- 5a + 3b - 2c + 7 (polynomial: four terms)
Exam Tip
Clearly distinguish between variables (letters) and constants (numbers). Variables can change; constants don't.
Common Mistake
Students confuse the variable with its coefficient. In 5x, the variable is x, but the coefficient (multiplier) is 5.
Like and Unlike Terms
Terms are parts of an expression separated by + or - signs.
Like Terms: Terms with the same variable(s) raised to the same power.
Examples: 3x and 5x (both have x¹), 2x² and -4x² (both have x²), 3xy and 7xy
Unlike Terms: Terms with different variables or different powers.
Examples: 3x and 3y (different variables), 2x and 2x² (different powers), 5a and 3b
Key Point: Only like terms can be combined. 3x + 5x = 8x, but 3x + 5y cannot be simplified further.
Exam Tip
Check powers carefully: 2x and 2x² are unlike terms and cannot be combined.
Common Mistake
Trying to combine unlike terms: 3a + 2b ≠ 5ab. They must be written separately.
Addition and Subtraction of Expressions
To add or subtract algebraic expressions, combine like terms only.
Example 1: Addition
(3x + 4) + (2x + 5)
= (3x + 2x) + (4 + 5) = 5x + 9
Example 2: Subtraction
(5x + 3) - (2x + 1)
= 5x + 3 - 2x - 1 = (5x - 2x) + (3 - 1) = 3x + 2
Important: When subtracting, distribute the negative sign to all terms in the second expression.
Exam Tip
Always arrange like terms together, then combine. It reduces errors.
Common Mistake
Forgetting to change signs when subtracting: (5x + 3) - (2x + 1) ≠ 5x + 3 - 2x + 1.
Multiplication and Division of Expressions
Multiplication by Monomial: Distribute the monomial to each term in the expression.
Example 1: 3 × (2x + 4) = 6x + 12
Example 2: x × (x + 3) = x² + 3x
Example 3: -2 × (3x - 5) = -6x + 10
Division by Monomial: Divide each term separately.
Example: (6x² + 9x) ÷ 3x = 2x + 3
Method: (6x²÷3x) + (9x÷3x) = 2x + 3
Exam Tip
Use the distributive property: a(b + c) = ab + ac. It works for multiplication and division.
Common Mistake
Missing terms when distributing: 2(x + 3) ≠ 2x (forgot to distribute to 3).
Simplification and Word Problems
Simplifying an expression means combining like terms and reducing to simplest form.
Example: Simplify 3x + 5 + 2x - 3
= (3x + 2x) + (5 - 3) = 5x + 2
Word Problems: Convert the problem into an algebraic expression, then solve.
- "A number x increased by 5" → x + 5
- "Twice a number x" → 2x
- "Three times x minus 4" → 3x - 4
- "Sum of x and y" → x + y
Exam Tip
In word problems, define variables clearly and check your answer makes sense in the context.
Common Mistake
Misinterpreting 'increased by' vs 'times': 'x increased by 5' is x + 5, NOT 5x.
Chapter Summary
Algebraic Expressions form the basis for equations and functions. Key topics:
- Variables & Constants: Letters represent unknowns; numbers are fixed values
- Terms & Classification: Monomials (1 term), Binomials (2), Polynomials (many)
- Like Terms: Same variable with same power; only like terms combine
- Operations: Add/subtract by combining like terms; multiply by distributing
- Simplification: Combine like terms to get simplest form
- Word Problems: Convert language to algebraic expressions
Exam Focus: Simplifying expressions, operations with polynomials, solving word problems, identifying like/unlike terms.