Algebra Lecture Notes

Introduction to Like Terms

• Like Terms: Terms with the same variable part.
  • Example: 5x + 4x are like terms.
  • Add coefficients: 5 + 4 = 9, so 5x + 4x = 9x.
• Unlike Terms: Terms with different variable parts.
  • Example: 3x + 4y + 5x + 8y.
  • Can only combine 3x + 5x = 8x and 4y + 8y = 12y.

Adding Radicals

• Like Radicals: Radicals with the same radicand.
  • Example: 3√2 + 5√7 + 8√2 + 3√7.
  • Combine: 3√2 + 8√2 = 11√2 and 5√7 + 3√7 = 8√7.

Adding Polynomials

• Combining Like Terms
  • Example: 7x + 4x² + 5x + 9x².
    • Combine like terms: 4x² + 9x² = 13x² and 7x + 5x = 12x.

Introduction to Polynomials

• Definitions:
  • Monomial: Single term (e.g., 8x).
  • Binomial: Two terms (e.g., 5x + 6).
  • Trinomial: Three terms (e.g., x² + 6x + 5).
  • Polynomial: Multiple terms.

Multiplying Polynomials

• Distributive Property: Multiply monomial by each term in another polynomial.
  • Example: 7x(x² + 2x - 3) = 7x³ + 14x² - 21x.
• FOIL Method: Multiply two binomials.
  • Example: (3x - 4)(2x + 7) results in 6x² + 13x - 28.

Exponent Rules

• Multiplication: Add exponents.
  • Example: x³ * x⁴ = x⁷.
• Division: Subtract exponents.
  • Example: x⁹ / x⁴ = x⁵.
• Power of a Power: Multiply exponents.
  • Example: (x²)³ = x⁶.
• Negative Exponents: Move base to opposite side of fraction.
  • Example: x⁻³ = 1/x³.

Solving Algebraic Equations

• Linear Equations:
  • Example: x + 4 = 9 -> x = 5.
  • Steps: Isolate variable using inverse operations.
• Quadratics:
  • Factoring, completing the square, or quadratic formula to find solutions.
  • Example: x² - 5x + 6 = 0 -> (x - 2)(x - 3).
• Rational Equations: Clear denominators, then solve.
  • Example: x/5 = 7/8 -> cross-multiply.

Graphing Linear Equations

• Slope-Intercept Form: y = mx + b.
  • m = slope, b = y-intercept.
  • Example: y = 2x - 1, slope = 2, intercept at (0, -1).
• Standard Form: Ax + By = C.
  • Find intercepts to graph.
  • Example: 2x - 3y = 6.

Writing Equations of Lines

• From Point and Slope: Use point-slope form, then convert to desired form.
• Given Two Points: Calculate slope, use point-slope form.
  • Example: Points (2, 4) and (-1, 5).
• Parallel/Perpendicular: Use slope relationships to determine new line.
  • Parallel: Same slope.
  • Perpendicular: Negative reciprocal slope.

Complex Fractions and Factorizations

• Factor by Grouping: Useful for polynomials with four terms.
  • Example: x³ - 4x² - x + 4 = (x - 4)(x² - 1).
• Complex Fractions: Simplify by keep-change-flip method for division.

Conclusion

• Practice problems provided for additional reinforcement.