Introduction to Algebra

• Algebra uses symbols and rules to manipulate those symbols.
• Symbols represent quantities without fixed values, called variables.
• Algebra is a unifying thread in almost all mathematics.

Basic algebraic operations

• Addition, subtraction, multiplication, and division are fundamental in algebra.
• Operations are performed on variables and constants.
• Constants have fixed numerical values.
• Algebraic expressions combine variables, constants, and operations.
• 3x + 5 is an algebraic expression, where x is a variable, and 3 and 5 are constants.

Variables and constants

• Variables, usually letters, represent unknown or changeable values.
• Constants are fixed numerical values that don't change.
• Coefficients are numerical or constant quantities multiplying the variable in an expression, e.g., 7 in 7y.
• In the expression 7y, 7 is the coefficient and y is the variable.

Expressions vs. equations

• An algebraic expression combines variables, constants, and operations, without an equals sign, like x^2 + 2y - 3.
• An algebraic equation states that two expressions are equal, indicated by an equals sign, like x + 5 = 10.

Solving equations

• Solving an equation means finding the value(s) of the variable(s) that make it true.
• The goal is to isolate the variable on one side of the equation.
• This is achieved by performing the same operations on both sides to maintain equality.

Linear equations

• A linear equation is one where the highest power of the variable is 1.
• The general form is ax + b = 0, where a and b are constants and x is the variable.
• Solving linear equations involves isolating the variable using inverse operations.

Quadratic equations

• A quadratic equation is one where the highest power of the variable is 2.
• The general form is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Factoring, completing the square, and the quadratic formula can solve quadratic equations.

Factoring quadratic equations

• Factoring involves expressing the quadratic expression as a product of two linear expressions.
• For example: x^2 + 5x + 6 = (x + 2)(x + 3)
• Setting each factor equal to zero determines the solutions.

Quadratic formula

• Solutions for x in a quadratic equation (ax^2 + bx + c = 0) are given by: x = (-b ± √(b^2 - 4ac)) / (2a)
• The discriminant (b^2 - 4ac) determines the nature of the roots.

Systems of equations

• A system of equations is a set of two or more equations with the same variables.
• Solving means finding values of the variables that satisfy all equations simultaneously.
• Solving methods include substitution, elimination, and graphing.

Substitution method

• This method solves one equation for one variable.
• Substitute that expression into the other equations.
• Solve the resulting equations for the remaining variables.
• Substitute the values back to find the values of the other variables.

Elimination method

• This method Multiplies equations by constants so that the coefficients of one variable are opposites.
• Add the equations to eliminate that variable.
• Solve for the remaining variable.
• Substitute the values back to find the values of the eliminated variable.

Inequalities

• Inequalities compare two expressions using symbols like <, >, ≤, ≥, ≠.
• < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), ≠ (not equal to)
• Solving inequalities involves finding values of the variable(s) that satisfy the inequality.

Properties of inequalities

• Adding or subtracting the same value from both sides does not change the inequality.
• Multiplying or dividing by a positive value does not change the inequality.
• Multiplying or dividing by a negative value reverses the inequality.

Polynomials

• Polynomials consist of variables and coefficients, involving addition, subtraction, and non-negative integer exponents.
• Example: 4x^3 - 2x^2 + x - 7

Operations with polynomials

• Polynomials can be added, subtracted, multiplied, and divided.
• Adding or subtracting involves combining like terms.
• Multiplying polynomials involves using the distributive property.

Exponents

• An exponent indicates how many times a base number is multiplied by itself, e.g., x^3 = x * x * x.

Rules of exponents

• Product of powers: x^a * x^b = x^(a+b)
• Quotient of powers: x^a / x^b = x^(a-b)
• Power of a power: (x^a)^b = x^(a*b)
• Zero exponent: x^0 = 1 (x ≠ 0)
• Negative exponent: x^(-a) = 1 / x^a

Radicals

• A radical (√) indicates the root of a number; the square root is most common.
• √9 = 3 because 3 * 3 = 9.

Simplifying radicals

• Simplifying radicals involves factoring out perfect squares, cubes, etc.
• Example: √20 = √(4 * 5) = √4 * √5 = 2√5

Rationalizing the denominator

• Eliminating radicals from the denominator of a fraction involves rationalization.
• Typically, multiply the numerator and denominator by the conjugate of the denominator to rationalize.

Logarithms

• Logarithms are the inverse operation to exponentiation; if b^y = x, then log_b(x) = y.
• b is the base
• x is the argument
• y is the exponent

Properties of logarithms

• Product rule: log_b(mn) = log_b(m) + log_b(n)
• Quotient rule: log_b(m/n) = log_b(m) - log_b(n)
• Power rule: log_b(m^p) = p * log_b(m)
• Change of base formula: log_b(a) = log_c(a) / log_c(b)

Functions

• A function relates inputs to permissible outputs, with each input related to exactly one output.
• The input is the argument, and the output is the value.

Function notation

• Functions are denoted by f(x), where x is the input variable.
• The output value of the function for a given input x is represented by f(x).

Domain and range

• The domain is the set of all possible input values (x) for which the function is defined.
• The range is the set of all possible output values (f(x)) that the function can produce.