Overview

This topic introduces how to interpret, manipulate, and simplify algebraic expressions. It covers operations involving polynomials, exponents, radicals, and algebraic fractions. A strong understanding of these skills forms the foundation for solving equations, factoring, and working with more complex expressions.

Key Concepts and Structures

• Algebraic Expressions: Combinations of numbers, variables, and operations. Examples: 3x + 2, 4a^2 - 5a + 1
• Like Terms: Terms with the same variable(s) raised to the same power. Combine by adding or subtracting coefficients.
• Distributive Property: Used to eliminate parentheses. a(b + c) = ab + ac
• Exponents: Rules include:
  • a^m * a^n = a^{m+n}
  • (a^m)^n = a^{mn}
  • a^0 = 1 (if a ≠ 0)
• Radicals: Simplify square roots and higher-order roots. Use rules like \sqrt{ab} = \sqrt{a} * \sqrt{b}.
• Factoring: Reverse of expanding. Includes factoring out the GCF, trinomials, and special products like a^2 - b^2 = (a - b)(a + b)
• Algebraic Fractions: Simplify by factoring and reducing common factors. Multiply/divide using reciprocal rules.
• Absolute Value: Distance from zero. Rules include:
  • |a| = a if a ≥ 0, |a| = -a if a < 0
  • |ab| = |a||b|

Step-by-Step Example

Problem: Simplify the expression 3(x + 2)^2 - 4x(x - 1)

Step 1: Expand each term.
(x + 2)^2 = x^2 + 4x + 4
3(x^2 + 4x + 4) = 3x^2 + 12x + 12
4x(x - 1) = 4x^2 - 4x

Step 2: Combine all parts:
3x^2 + 12x + 12 - (4x^2 - 4x)

Step 3: Distribute the negative and combine like terms:
3x^2 + 12x + 12 - 4x^2 + 4x = -x^2 + 16x + 12

Final Answer: -x^2 + 16x + 12

Quick Tip