Introduction to Algebra

• Algebra involves symbols and rules to manipulate them.
• Symbols in algebra represent quantities without fixed values, known as variables.
• Algebra includes solving equations to discover variable values satisfying conditions.
• Algebra extends arithmetic using symbols to represent numbers.

Expressions

• Algebraic expressions combine variables, numbers, and arithmetic operations.
• 3x + 2, y^2 - 5, and 2ab + c are examples of algebraic expressions.
• Expressions do not include an equals sign (=).

Variables

• Variables are symbols, usually letters, representing unknown or changing quantities.
• x is the variable in the expression 3x + 2.
• Variables can assume different values.

Constants

• Constants are fixed values that do not change.
• 2 is the constant in the expression 3x + 2.

Coefficients

• A coefficient is a number multiplying a variable.
• 3 is the coefficient of x in the expression 3x + 2.
• A variable without a coefficient has an implied coefficient of 1 (e.g., x is the same as 1x).

Terms

• Terms are individual parts of an algebraic expression, separated by addition or subtraction.
• The terms in 3x + 2 are 3x and 2.
• The terms in y^2 - 5 are y^2 and -5.

Like Terms

• Like terms have the same variable raised to the same power.
• 3x and 5x are like terms.
• 2y^2 and -4y^2 are like terms.
• 3x and 3x^2 are not like terms because the powers of x differ.
• 2x and 2y are not like terms because the variables are different.

Combining Like Terms

• Like terms combine through addition or subtraction of their coefficients.
• 3x + 5x = 8x
• 2y^2 - 4y^2 = -2y^2

Order of Operations

• PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) dictates the order of operations.
• Operations inside parentheses or brackets are performed first.
• Exponents or powers are evaluated next.
• Multiplication and division are then performed from left to right.
• Finally, addition and subtraction are performed from left to right.

Equations

• An equation states that two expressions are equal.
• Equations contain an equals sign (=).
• Examples of equations: 3x + 2 = 5, y^2 - 1 = 0, and 2a = b + c.

Solving Equations

• Solving an equation means finding the value(s) of the variable(s) that make the equation true.
• It typically involves isolating the variable on one side of the equation.
• The goal is to isolate the variable on one side of the equals sign.

Addition and Subtraction Properties of Equality

• Adding the same number to both sides does not change the solution; if a = b, then a + c = b + c.
• Subtracting the same number from both sides does not change the solution; if a = b, then a - c = b - c.

Multiplication and Division Properties of Equality

• Multiplying both sides by the same non-zero number does not change the solution; if a = b, then ac = bc.
• Dividing both sides by the same non-zero number does not change the solution; if a = b, then a/c = b/c (where c ≠ 0).

Linear Equations

• A linear equation has a highest variable power of 1.
• Linear equations can be written as ax + b = c, where a, b, and c are constants, and x is the variable.
• Solving a linear equation involves isolating the variable using inverse operations.

Example of Solving a Linear Equation

• Solve the equation 3x + 2 = 5.
• Subtract 2 from both sides: 3x + 2 - 2 = 5 - 2, simplifying to 3x = 3.
• Divide both sides by 3: 3x / 3 = 3 / 3, simplifying to x = 1.
• The equation's solution is therefore x = 1.

Quadratic Equations

• Quadratic equations have a highest variable power of 2.
• Quadratic equations can be written as ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Factoring, completing the square, or the quadratic formula can solve quadratic equations.

Factoring Quadratic Equations

• Factoring involves expressing the quadratic expression as a product of two binomials.
• For example, x^2 + 5x + 6 = (x + 2)(x + 3).
• If (x + 2)(x + 3) = 0, then either x + 2 = 0 or x + 3 = 0, implying x = -2 or x = -3.

Quadratic Formula

• The quadratic formula generally solves any quadratic equation.
• For ax^2 + bx + c = 0, the solutions are: x = (-b ± √(b^2 - 4ac)) / (2a).

Systems of Equations

• A system of equations is two or more equations containing the same variables.
• Systems of equations are solved to find variable values that satisfy all equations simultaneously.
• Substitution and elimination are common solution methods.

Substitution Method

• Solve one equation for one variable, then substitute that expression into the other equation(s).
• The resulting equation(s) can then be solved for the remaining variable(s).

Elimination Method

• Manipulate equations so that one variable's coefficients are opposites.
• Adding the equations eliminates that variable.
• Solve the resulting equation for the remaining variable.

Inequalities

• An inequality compares two expressions using symbols like <, >, ≤, or ≥.
• Examples of inequalities include: x + 3 < 5, 2y - 1 ≥ 7, and a < b + c.

Solving Inequalities

• Solving an inequality finds the variable values that make the inequality true.
• The rules for solving inequalities mirror those for equations, with an exception.
• Multiplying or dividing by a negative number reverses the inequality sign's direction.

Example of Solving an Inequality

• Solve the inequality -2x + 4 < 10.
• Subtract 4 from both sides: -2x < 6.
• Divide both sides by -2 (and reverse the inequality sign): x > -3.
• Therefore, the inequality's solution is x > -3.

Functions

• A function relates inputs to permissible outputs, where each input links to exactly one output.
• Functions can be represented by equations, graphs, or tables.
• The input is often x, and the output is often y or f(x).

Function Notation

• Function notation uses f(x) to write functions.
• f(x) is the output of function f when the input is x.
• For instance, if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.

Domain and Range

• A function's domain includes all possible input values (x-values).
• A function's range includes all possible output values (y-values).

Exponents

• Exponents indicate how many times a base number is multiplied by itself.
• x^n means x multiplied by itself n times.
• For example, 2^3 = 2 * 2 * 2 = 8.

Laws of Exponents

• Product of Powers: x^m * x^n = x^(m+n)
• Quotient of Powers: x^m / x^n = x^(m-n)
• Power of a Power: (x^m)^n = x^(m*n)
• Power of a Product: (xy)^n = x^n * y^n
• Power of a Quotient: (x/y)^n = x^n / y^n
• Zero Exponent: x^0 = 1 (for x ≠ 0)
• Negative Exponent: x^(-n) = 1 / x^n

Polynomials

• A polynomial is an algebraic expression of variables and coefficients with addition, subtraction, multiplication, and non-negative integer exponents.
• Examples of polynomials include: x^2 + 3x + 2, 4y^3 - 2y + 1, and 5.

Types of Polynomials

• Monomial: One term (e.g., 5x^2).
• Binomial: Two terms (e.g., 2x + 3).
• Trinomial: Three terms (e.g., x^2 + 3x + 2).

Operations with Polynomials

• Polynomials support addition, subtraction, multiplication, and division.
• Addition and subtraction combine like terms.
• Multiplication uses the distributive property.
• Division uses long division or synthetic division.

Factoring Polynomials

• Factoring a polynomial expresses it as a product of simpler polynomials.
• Factoring reverses multiplication.
• Common techniques include factoring out the GCF, grouping, and factoring quadratic trinomials.