Algebra Class: Simplifying Expressions

Simplifying Expressions

• Simplifying expressions involves rewriting an expression in a simpler form while maintaining the same value.
• This typically means combining like terms, applying order of operations (PEMDAS/BODMAS), and reducing fractions.
• The goal is to make expressions easier to understand, work with, and evaluate.

Combining Like Terms

• Like terms are terms that contain the same variables raised to the same powers.
• Only like terms can be combined.
• To combine like terms, add or subtract the coefficients of the terms and keep the variables and exponents the same.
• Example: 3x + 5x = 8x; 2x² - x² = x²
• Example: 2a + 3b - a + 4b simplifies to a + 7b.

Order of Operations (PEMDAS/BODMAS)

• PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) specify the order of operations in expressions with multiple operations.
• Parentheses/Brackets: Evaluate expressions inside parentheses first.
• Exponents/Orders: Evaluate exponents after parentheses.
• Multiplication and Division (from left to right): Perform multiplication or division operations in order from left to right.
• Addition and Subtraction (from left to right): Perform addition or subtraction operations in order from left to right.
• Example: 2 + 3 * 4 = 2 + 12 = 14 (Multiplication done before addition).
• Example: 10 / 2 + 3 = 5 + 3 = 8

Simplifying Fractions

• Simplifying fractions involves reducing a fraction to its lowest terms.
• To simplify a fraction, divide both the numerator and denominator by their greatest common factor (GCF).
• Example: 10/15 simplifies to 2/3 because the GCF of 10 and 15 is 5.

Simplifying Algebraic Expressions

• Simplifying algebraic expressions follows the same principles, combining like terms and evaluating operations in the correct order.
• Expressions within parentheses require following the order of operations.
• Example: 2(x + 3) + 4x simplifies to 6x + 6.

Factoring

• Factoring expresses an expression as a product of its factors.
• Factoring is the opposite of expanding.
• Factoring is helpful for simplifying expressions, solving equations, and other mathematical applications.
• Example: x² + 2x factors to x(x + 2).

Distributing

• Distributing involves multiplying a single term by each term inside parentheses or similar groupings.
• Example: 3(x + 2) = 3x + 6

Combining radicals

• Combine only like radicals (radicals with the same radicand and index).
• To combine like radicals, add or subtract their coefficients, keeping the radicand the same.
• Example: 2√(3) + 5√(3) = 7√(3)

Solving Equations Involving Simplification

• Solving equations involving simplification often requires simplifying the equation first, then applying standard equation solving methods (isolating the variable).
• Example: To solve 2x + 5 = 9, simplify by subtracting 5 and then divide by 2.
• Example: The solution to 2(x + 3) = 10 is x = 2.

Important Concepts:

• Practice problems are vital for applying and understanding the concepts.
• Use various examples to review techniques in various scenarios.
• Mastery takes commitment and focused effort.