Algebraic Expressions - Formulas, Simplifying, Evaluating

Introduction to Algebraic Expressions

• Algebraic expressions are mathematical statements formed when operations like addition, subtraction, multiplication, division, etc., are performed on variables and constants.
• Example: To form a pattern with "4" using matchsticks:
  • Pattern: 4 + 3(n-1)
  • This is an algebraic expression.

Components of Algebraic Expressions

Variables, Constants, Terms, and Coefficients

• Variable: A symbol with no fixed value (e.g., n, x, y).
• Constant: A symbol with a fixed numerical value (e.g., 3, 5).
• Term: Can be a variable, a constant, or a combination of both through multiplication or division (e.g., 3x², 5).
• Coefficient: A number multiplying a variable in a term.

Simplifying Algebraic Expressions

• Combine like terms and variables with the same powers.
• Example: Simplify x³ + 3x² - 2x³ + 2x - x² + 3 + x to x³ + 2x² + x + 3.

Operations on Algebraic Expressions

• Addition: Combine like terms.
  • Example: (x² + 2x + 3) + (2x² - 3x) = 3x² - x + 3
• Subtraction: Add the additive inverse of the second expression.
  • Example: (3x² - 5x) - (x² - 2x + 2) = 2x² - 3x - 2
• Multiplication: Multiply each term of one expression by each term of the other.
  • Example: ab(2ab + 3) = 2a²b² + 3ab
• Division: Factor the numerator and denominator, cancel terms.
  • Example: 2x² / (2x² + 4x) = x / (x + 2)

Algebraic Expression Formulas

• Formulas simplify solving equations and memorizing expressions.
  • (a + b)² = a² + 2ab + b²
  • (a - b)² = a² - 2ab + b²
  • (a + b)(a - b) = a² - b²

Types of Algebraic Expressions

• Monomial: One term with non-negative integer exponents (e.g., 3xy).
• Binomial: Two monomials (e.g., (3/4)x - 2y²).
• Trinomial: Three monomials (e.g., 3x - 2y + z).
• Polynomial: One or more monomials (e.g., -2/3x³ + 7x² + 3x + 5).
• Multinomial: One or more terms with positive or negative exponents (e.g., 4x⁻¹ + 2y + 3z).

Examples of Algebraic Expressions

1. Example 1: If there are 25 oranges in a bag, the expression for oranges in x bags is 25x.
2. Example 2: The expression 4x + 5 is a binomial and a polynomial.
3. Example 3: 12y/x is not a monomial because the denominator is a variable.
4. Example 4: Sum of 3x + 2 and 4y + 2z is 3x + 4y + 2z + 2.