Algebra Lecture Notes

Key Concepts Covered

• Like Terms
• Combining Like Terms
• Polynomials
• Multiplying Monomials, Binomials & Trinomials
• Properties of Exponents
• Solving Equations
• Factoring
• Graphing Linear Equations
• Writing Equations of Lines

Like Terms

• Definition: Terms with the same variable and exponent.
• Example: 5x and 4x are like terms because both have 'x'.
• Combining: Add coefficients: 5x + 4x = 9x.

Combining Like Terms

• Example: 3x + 4y + 5x + 8y
  • Combine like terms: 3x + 5x = 8x, 4y + 8y = 12y.

Polynomials

• Monomial: Single term (e.g., 8x).
• Binomial: Two terms (e.g., 5x + 6).
• Trinomial: Three terms (e.g., x^2 + 6x + 5).
• Polynomial: Many terms.

Multiplying Terms

• Monomial & Trinomial: Distribute the monomial across the trinomial.
• Example: 7x * (x^2 + 2x - 3)
  • Multiply each term: 7x * x^2 = 7x^3, 7x * 2x = 14x^2, 7x * -3 = -21x.

Properties of Exponents

• Multiplying: Add exponents: x^a * x^b = x^(a+b).
• Dividing: Subtract exponents: x^a / x^b = x^(a-b).
• Power of Power: Multiply exponents: (x^a)^b = x^(a*b).

Solving Equations

• Basic Steps:
  1. Isolate the variable.
  2. Use inverse operations (addition/subtraction, multiplication/division).
• Example:
  • 3x + 5 = 11
  • Subtract 5: 3x = 6
  • Divide by 3: x = 2

Factoring

• Factoring Out GCF: Greatest Common Factor.
• Difference of Squares: a^2 - b^2 = (a + b)(a - b).
• Trinomial Factoring: Find two numbers that multiply to the constant term and add to the linear coefficient.

Graphing Linear Equations

• Slope-Intercept Form: y = mx + b
  • m = slope (rise/run)
  • b = y-intercept
• Standard Form: Ax + By = C
  • Useful for finding x and y intercepts.

Writing Equations of Lines

• Point-Slope Form: y - y1 = m(x - x1)
• Slope-Intercept Form: Convert from point-slope form.
• Parallel & Perpendicular Lines:
  • Parallel: Same slope.
  • Perpendicular: Negative reciprocal slope.