Algebra: The Basics

• Algebra is a mathematical tongue that doth traffic in symbols and the governance of their wielding.
• These tokens do stand for quantities without settled worth, and are named variables.
• Algebra is of a broader kind than arithmetic, which handleth only with numbers of certain value.

Basic Algebra Wit

• Variables: Tokens, oft letters, that betoken quantities unknown or yet changing. In the saying 3x + 2, x is such a variable.
• Constants: Values steadfast that do not shift. In 3x + 2, 3 and 2 are constants.
• Coefficients: Numbers that do multiply variables. In 3x + 2, 3 be the coefficient of x.
• Expressions: Gatherings of variables, constants, and mathematical deeds. 3x + 2 and x^2 - 5x + 6 be showings of algebra expressions.
• Equations: Declarations that twain expressions be equal. Equations hold an equals sign (=). For example, 3x + 2 = 11.
• Terms: Parts of an expression set off by addition or subtraction. In 3x + 2, 3x and 2 be terms.
• Operators: Tokens that show mathematical deeds, such as addition (+), subtraction (-), multiplication (*), division (/), and raising to a power (^).

Foundational Werk

• Addition and Subtraction: Joining like terms. Like terms share the same variable raised to the same power. For example, 3x + 2x = 5x.
• Multiplication: Spreading a term across an expression. For example, 2(x + 3) = 2x + 6.
• Division: Sundering each term in an expression by a shared factor. For example, (4x + 6) / 2 = 2x + 3.
• Exponents: Signifying repeated multiplication. For example, x^3 = x * x * x.

Solving Equations

• To solve an equation is to find the worth of the variable(s) that maketh the equation true.
• Linear Equations: Equations where the highest power of the variable is 1. Example: ax + b = 0.
  • To solve, set apart the variable by doing the opposite deeds on both sides of the equation.
  • Example: 2x + 3 = 7. Subtract 3 from both sides: 2x = 4. Divide both sides by 2: x = 2.
• Quadratic Equations: Equations where the highest power of the variable is 2. General form: ax^2 + bx + c = 0.
  • Factoring: Uttering the quadratic expression as a yield of twain linear factors. Set each factor to zero and solve for x. Example: x^2 - 5x + 6 = (x - 2)(x - 3) = 0. Therefore, x = 2 or x = 3.
  • Quadratic Formula: Used when factoring is hard or cannot be done. The formula is: x = (-b ± √(b^2 - 4ac)) / (2a).
  • Completing the Square: A way to write the quadratic equation in a guise that allows easy solving.
• Systems of Equations: A set of two or more equations with the same variables.
  • Substitution: Solve one equation for one variable and put that saying into the other equation.
  • Elimination: Multiply one or both equations by a constant so that the coefficients of one variable are opposites. Then add the equations together to do away with that variable.

Unequalities

• Inequalities do compare twain expressions using tokens such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
• Solving inequalities is like solving equations, save for one weighty thing: multiplying or dividing by a negative number doth turn the inequality sign about.
• Example: 3x + 2 < 8. Subtract 2 from both sides: 3x < 6. Divide both sides by 3: x < 2. The answer is all values of x less than 2.
• Interval Notation: A way to show sets of numbers. For example, x < 2 is writ as (-∞, 2). Curves show that the ending is not held, while brackets show that it is.

Polynomials

• Polynomials be expressions made of variables and coefficients, joined by addition, subtraction, and non-negative whole number powers.
• Examples: x^2 + 3x - 4, 5x^4 - 2x + 1, 7.
• Degree of a Polynomial: The highest power of the variable in the polynomial.
• Types of Polynomials:
  • Monomial: A polynomial with one term (e.g., 5x^2).
  • Binomial: A polynomial with two terms (e.g., x + 3).
  • Trinomial: A polynomial with three terms (e.g., x^2 + 2x + 1).
• Operations with Polynomials:
  • Addition and Subtraction: Join like terms.
  • Multiplication: Use the spreading law.
  • Division: Polynomial long division or made-up division.
• Factoring Polynomials: Uttering a polynomial as a yield of simpler polynomials.

Functions

• A function is a tie between a set of inputs and a set of allowed outputs with the trait that each input is tied to only one output.
• Showing: Functions be oft marked by f(x), where x is the input and f(x) is the output.
• Domain: The set of all mayhap input values for a function.
• Range: The set of all mayhap output values for a function.
• Types of Functions:
  • Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
  • Quadratic Functions: f(x) = ax^2 + bx + c.
  • Polynomial Functions: Functions defined by polynomials.
  • Rational Functions: Functions that be rates of polynomials.
  • Exponential Functions: f(x) = a^x, where a is a constant.
  • Logarithmic Functions: The reverse of exponential functions.
• Graphing Functions: A seeing show of a function on a coordinate plane.

Powers and Roots

• Powers: Show repeated multiplying. x^n doth mean x multiplied by itself n times.
  • Laws of Powers:
    • x^m * x^n = x^(m+n)
    • x^m / x^n = x^(m-n)
    • (x^m)^n = x^(m*n)
    • (xy)^n = x^n * y^n
    • (x/y)^n = x^n / y^n
    • x^0 = 1
    • x^(-n) = 1 / x^n
• Roots: The reverse deed of raising to a power. The nth root of x is writ as √[n](x).
  • Easing Roots: Finding the biggest perfect nth power that parts x.
  • Clearing the Denominator: Taking roots from the denominator of a fraction.

Logs

• A logarithm be the reverse deed to raising to a power. log_b(x) = y means b^y = x.
• Base: The base of the logarithm (b in log_b(x)). Common bases are 10 (common logarithm) and e (natural logarithm, marked ln(x)).
• Qualities of Logs:
  • log_b(xy) = log_b(x) + log_b(y)
  • log_b(x/y) = log_b(x) - log_b(y)
  • log_b(x^n) = n * log_b(x)
  • log_b(1) = 0
  • log_b(b) = 1
  • Change of Base Formula: log_a(x) = log_b(x) / log_b(a)
• Solving Log Equations: Use the qualities of logs to set apart the variable.

Absolute Worth

• The absolute worth of a number is its step from zero. Marked by |x|.
• |x| = x if x ≥ 0, and |x| = -x if x < 0.
• Solving Absolute Worth Equations: Think on both plus and minus cases.
  • For example, if |x - 2| = 3, then x - 2 = 3 or x - 2 = -3. Solving these gives x = 5 or x = -1.
• Solving Absolute Worth Inequalities:
  • If |x| < a, then -a < x < a.
  • If |x| > a, then x < -a or x > a.