Common Algebra Concepts

May 18, 2024

Key Concepts in Algebra

Like Terms

  • Definition: Terms with the same variable(s) raised to the same power(s).
  • Example:
    • 5x + 4x are like terms. Add coefficients: 5 + 4 = 9, so 5x + 4x = 9x.
    • 3x + 4y + 5x + 8y: Combine like terms, 3x + 5x = 8x and 4y + 8y = 12y.
    • Radical terms: 3√2 + 8√2 = 11√2, and 5√7 + 3√7 = 8√7.
  • Complex Example: 7x + 4x^2 + 5x + 9x^2: Combine like terms, 4x^2 + 9x^2 = 13x^2 and 7x + 5x = 12x.

Polynomials

  • Monomial: A single term (e.g., 8x, 5x^2).
  • Binomial: Two terms (e.g., 5x + 6).
  • Trinomial: Three terms (e.g., x^2 + 6x + 5).
  • Polynomial: An expression with many terms.

Adding and Subtracting Polynomials

  • Combine like terms by adding or subtracting their coefficients.
  • Example: 9x^2 + 6x + 5 + 3x^2 - 5x - 9: 9x^2 + 3x^2 = 12x^2, 6x - 5x = x, and 5 - 9 = -4.
  • Distributive Property: Distribute the negative sign or coefficient across terms within parentheses.

Multiplying Terms

  • Monomial x Trinomial: Distribute the monomial to each term of the trinomial.
    • Example: 7x * (x^2 + 2x - 3) = 7x^3 + 14x^2 - 21x.
  • Binomial x Binomial (FOIL Method): First, Outer, Inner, Last terms are multiplied and then combined.
    • Example: (3x - 4)(2x + 7) → FOIL → 6x^2 + 21x - 8x - 28 = 6x^2 + 13x - 28.

Exponent Rules

  • Multiplication: Add exponents for the same base: x^3 * x^4 = x^7.
  • Division: Subtract exponents for the same base: x^9 / x^4 = x^5.
  • Power to a Power: Multiply the exponents: (x^2)^3 = x^6.
  • Negative Exponents: x^-3 = 1/x^3.
    • Example: 3x^4 y^5 * 5x^6 y^7 = 15x^10 y^12.

Special Exponents and Roots

  • Distributive Property: Apply the exponent to both the coefficient and the variable part when raised to a power.
    • Example: (4x^2 y^3)^3 = 4^3 * x^6 * y^9 = 64x^6 y^9.
  • Zero Exponent: Any term raised to the zero power is 1 (except 0^0 which is undefined).

Solving Equations

  • Basic Approach: Isolate the variable using inverse operations.
    • Example: x + 4 = 9 → Subtract 4 from both sides → x = 5.
    • 3x + 5 = 11 → Subtract 5, divide by 3 → x = 2.
  • Fractions: Clear fractions by multiplying through by the common denominator.
    • Example: (2/3)x + 5 = 8 → Multiply by 3 → 2x + 15 = 24 → x = 3/2.

Factoring Quadratics

  • Difference of Squares: a^2 - b^2 = (a - b)(a + b).
  • Trinomials with Leading Coefficient of One: x^2 + bx + c. Find two numbers that multiply to c and add to b.
    • Example: x^2 - 5x + 6 → (x - 2)(x - 3).
  • Trinomials with a Leading Coefficient Other Than One: Use the AC method or quadratic formula.
    • Example: 2x^2 + 3x - 2 → Factor by grouping.

Quadratic Formula

  • Quadratic formula: x = [-b ± √(b^2 - 4ac)] / 2a.
  • Use: When factoring is difficult or impractical.
    • Example: 6x^2 + 7x - 3 → Use quadratic formula.

Graphing Linear Equations

  • Slope-Intercept Form: y = mx + b.
  • Plotting:
    • Start with the y-intercept (b).
    • Use the slope (m) to find another point.
    • Draw the line.
  • Standard Form: Ax + By = C. Find x- and y-intercepts to plot.

Writing Equations of Lines

  • Given Point and Slope: Use point-slope form: y - y1 = m(x - x1).
  • Converting Forms: Convert between point-slope, slope-intercept, and standard forms as needed.
  • Parallel and Perpendicular Lines:
    • Parallel lines have the same slope.
    • Perpendicular lines have slopes that are negative reciprocals.
    • Example: Line through (1, 3) parallel to 2x - 3y - 5 = 0. Convert to slope-intercept form, find slope, and use point-slope form for new line.