Basic Algebra Concepts

Study Notes

Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in equations and formulas. It provides a general framework for solving problems involving unknowns.
Variables (represented by letters like 'x', 'y', 'z') are used to denote unknown values. Constants are fixed values.
Equations express relationships between variables and constants. They typically involve an equals sign (=).
Inequalities express a relationship of greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) between expressions.
Expressions combine variables, constants, and mathematical operations (addition, subtraction, multiplication, division, exponents, etc.).

Adding and Subtracting: Like terms (terms with the same variable raised to the same power) are combined. Unlike terms are left as they are.
Multiplying: The product of coefficients and the sum of exponents of like bases determine the result.
Dividing: The quotient of coefficients and the difference of exponents of like bases are involved.

The goal is to isolate the variable on one side of the equation.
Addition, subtraction, multiplication, and division are the fundamental operations used to isolate the variable.
Inverse operations are applied to both sides of the equation.
Solving for a variable usually involves multiple steps to isolate it.

Linear equations involve variables raised to the power of 1 (e.g., y = mx + b).
Their graphs are straight lines.
The slope (m) describes the steepness of the line, and the y-intercept (b) is the point where the line crosses the y-axis.
Systems of linear equations can be solved graphically, using substitution, or using elimination methods.

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The variables are raised to non-negative integer powers.
Degree of a polynomial: The highest power of the variable in the polynomial.
Types of polynomials include monomials (one term), binomials (two terms), trinomials (three terms), and so on.
Operations like addition, subtraction, multiplication, and division can be performed with polynomials following the rules of algebra.

Factoring is the process of rewriting an expression as a product of its factors.
This can help simplify expressions, solve equations, and perform other operations more easily.
Various methods exist for factoring, including taking out common factors, grouping, difference of squares, and the quadratic formula.

Quadratic equations involve variables raised to the power of 2 (e.g., ax² + bx + c = 0).
The solutions to quadratic equations can be found using the quadratic formula, factoring, or completing the square.
The graph of a quadratic equation is a parabola.

Exponents represent repeated multiplication, with the base raised to a power (e.g., x²).
Radicals (square roots, cube roots, etc.) are used to represent inverse operations of exponents.
Properties of exponents and radicals provide tools for simplifying and operating on expressions containing them.