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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.
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