Learning Objectives
- Understand terms, factors, and coefficients of algebraic expressions
- Classify expressions as monomials, binomials, trinomials, and polynomials
- Perform addition, subtraction, and multiplication of algebraic expressions
- Learn and apply standard algebraic identities
Key Concepts
Algebraic Expressions
An algebraic expression is a combination of constants and variables connected by +, -, ×, and ÷. Each part separated by + or - is called a term. The numerical part of a term is its coefficient.
Like terms: Terms with the same variables raised to the same powers (e.g., 3xy and -5xy).
Unlike terms: Terms with different variables or different powers (e.g., 3xy and 3x²y).
Types of Expressions
- Monomial: Expression with one term (e.g., 5x, -3ab²)
- Binomial: Expression with two terms (e.g., 3x + 2)
- Trinomial: Expression with three terms (e.g., x² + 2x + 1)
- Polynomial: Expression with one or more terms where variables have whole number exponents
Operations on Algebraic Expressions
Addition: Combine like terms by adding their coefficients.
Subtraction: Change the sign of each term in the expression being subtracted, then add.
Multiplication: Multiply each term of one expression with every term of the other. Use the rule: xᵃ × xᵇ = xᵃ⁺ᵇ.
Standard Algebraic Identities
An identity is an equation true for all values of the variables.
- Identity I: (a + b)² = a² + 2ab + b²
- Identity II: (a - b)² = a² - 2ab + b²
- Identity III: (a + b)(a - b) = a² - b²
- Identity IV: (x + a)(x + b) = x² + (a + b)x + ab
Applications of Identities
Identities are used for quick multiplication of numbers, simplifying expressions, and factorisation.
Example: 105² = (100 + 5)² = 100² + 2(100)(5) + 5² = 10000 + 1000 + 25 = 11025.
Summary
Algebraic expressions consist of terms with variables and coefficients. They are classified by the number of terms. Operations follow the rules of combining like terms and exponent laws. Standard identities provide shortcuts for expanding and simplifying expressions.
Important Terms
- Term: A part of an expression separated by + or - signs
- Coefficient: The numerical factor of a term
- Like Terms: Terms with identical variable parts
- Polynomial: An expression with whole number exponents on variables
- Identity: An equation that holds true for all values of its variables
Quick Revision
- Like terms can be added/subtracted; unlike terms cannot
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- (a + b)(a - b) = a² - b²
- (x + a)(x + b) = x² + (a+b)x + ab
- Use identities for quick calculation of squares and products