Algebra Lecture: Key Concepts

Like Terms and Combining

• Like Terms: Terms with the same variable raised to the same power.
  • Example: 5x + 4x are like terms.
  • Combine by adding coefficients: 5x + 4x = 9x.
• Non-Like Terms: Cannot be directly added.
  • Example: 3x + 4y + 5x + 8y.
  • Combine like terms: 3x + 5x = 8x and 4y + 8y = 12y.

Radicals

• Combining Radicals: Only like radicals can be combined.
  • Example: 3√2 + 8√2 = 11√2.
  • 5√7 + 3√7 = 8√7.

Exponents and Polynomials

• Polynomial Types:
  • Monomial: Single term (e.g., 5x).
  • Binomial: Two terms (e.g., x + 3).
  • Trinomial: Three terms (e.g., x^2 + 5x + 6).
  • Polynomial: Many terms.
• Multiplying Monomials:
  • Distribute each term: 7x(x^2 + 2x - 3).
• FOIL Method for Binomials:
  • First, Outer, Inner, Last: (3x - 4)(2x + 7).

Exponent Rules

• Product of Powers: Add exponents (x^a * x^b = x^(a+b)).
• Quotient of Powers: Subtract exponents (x^a / x^b = x^(a-b)).
• Power of a Power: Multiply exponents ((x^a)^b = x^(a*b)).
• Zero Exponent: Any term raised to 0 is 1.

Solving Linear Equations

• Simple Equations: Solve by isolating x.
  • Example: x + 4 = 9, solve by subtracting 4.
• Multi-Step Equations: Distribute first, then combine like terms.
  • Example: 5 - 3(x + 4) = 7 + 2(x - 1).

Quadratic Equations

• Factoring: Use when the quadratic can be expressed as (x + a)(x + b) = 0.
• Quadratic Formula: Use when factoring is difficult.
  • Formula: x = [-b ± √(b^2 - 4ac)] / 2a.

Graphing Linear Equations

• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
• Standard Form: Ax + By = C.
  • Find x- and y-intercepts for graphing.

Writing Equations

• From Two Points: Find slope first, then use point-slope form.
• Parallel and Perpendicular Lines:
  • Parallel: Same slope.
  • Perpendicular: Negative reciprocal slope.

Practice Problems

• Work on simplifying expressions, solving equations, and graphing lines using the methods and concepts above.