Algebra: Variables, Constants and Expressions

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols
• It is a unifying thread of almost all of mathematics

Variables

• A variable is a symbol (usually a letter) that represents a value that is unknown or can change
• Variables are used to express relationships between quantities
• In the expression 3x + 5, x is a variable
• In equations or inequalities a variable is an unknown that is solved to find its value
• In functions, variables represent an input to the function

Constants

• A constant is a value that does not change
• Constants can be numbers, or symbols that are defined to have a fixed value
• In the expression 3x + 5, 3 and 5 are constants

Expressions

• An algebraic expression is a combination of variables, constants, and operations (addition, subtraction, multiplication, division, exponentiation, etc.)
• Expressions do not contain an equals sign
• 3x + 5, a^2 - b^2, and sqrt(y) are expressions

Equations

• An equation is a statement that two expressions are equal
• Equations contain an equals sign (=)
• 3x + 5 = 14 is an equation

Inequalities

• An inequality is a statement that compares two expressions using inequality symbols like < (less than), > (greater than), <= (less than or equal to), or >= (greater than or equal to)
• 3x + 5 < 14 is an example of an inequality

Operations

• Basic algebraic operations include addition, subtraction, multiplication, division, and exponentiation
• PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is an acronym that helps to remember the order of operations

Terms

• Terms are the individual parts of an expression or equation, separated by addition or subtraction
• In the expression 3x + 5, 3x and 5 are terms

Coefficients

• A coefficient is a number that is multiplied by a variable
• In the term 3x, 3 is the coefficient of x

Like Terms

• Like terms are terms that have the same variable raised to the same power
• Like terms can be combined by adding or subtracting their coefficients
• 3x and 5x are like terms and can be combined to give 8x

Polynomials

• A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents
• 3x^2 + 2x - 1 is a polynomial

Monomial

• A monomial is a polynomial with only one term
• 5x^2 is a monomial

Binomial

• A binomial is a polynomial with two terms
• 2x + 3 is a binomial

Trinomial

• A trinomial is a polynomial with three terms
• x^2 + 2x + 1 is a trinomial

Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable in the polynomial
• The degree of 3x^2 + 2x - 1 is 2

Solving Equations

• Solving an equation involves finding the value(s) of the variable(s) that make the equation true
• This is typically achieved by isolating the variable on one side of the equation by performing the same operations to both sides of the equation

Linear Equations

• A linear equation is an equation in which the highest power of the variable is 1
• Linear equations can be written in the form ax + b = 0, where a and b are constants and x is the variable

Solving Linear Equations

• To solve a linear equation, isolate the variable by performing inverse operations
• To solve 3x + 5 = 14, subtract 5 from both sides to get 3x = 9, then divide both sides by 3 to get x = 3

Quadratic Equations

• A quadratic equation is an equation in which the highest power of the variable is 2
• Quadratic equations can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable

Solving Quadratic Equations

• Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula
• Quadratic Formula: For a quadratic equation ax^2 + bx + c = 0, the solutions are given by x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Factoring

• Factoring involves expressing a polynomial as a product of two or more simpler polynomials
• x^2 - 4 can be factored as (x + 2)(x - 2)

Completing the Square

• Completing the square is a technique used to convert a quadratic equation into a perfect square trinomial, which can then be easily solved

Systems of Equations

• A system of equations is a set of two or more equations containing the same variables
• The solution to a system of equations is the set of values for the variables that make all the equations true

Solving Systems of Equations

• Systems of equations can be solved by substitution or elimination

Substitution

• Substitution involves solving one equation for one variable and then substituting that expression into the other equation

Elimination

• Elimination involves adding or subtracting the equations in a system to eliminate one of the variables

Inequalities

• Inequalities are mathematical statements that compare two expressions using inequality symbols
• Less than is represented by: <
• Greater than is represented by: >
• Less than or equal to is represented by: <=
• Greater than or equal to is represented by: >=

Solving Inequalities

• Solving inequalities is similar to solving equations, but with one important difference: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed

Absolute Value

• The absolute value of a number is its distance from zero on the number line
• The absolute value of a number is always non-negative
• Absolute value is denoted by |x|
• |3| = 3 and |-3| = 3

Properties of Absolute Value

• |x| >= 0 for all x
• |-x| = |x| for all x
• |xy| = |x||y| for all x and y

Radicals

• A radical is a mathematical expression that involves a root, such as a square root, cube root, etc.
• sqrt(x) denotes the most common radical, the square root

Simplifying Radicals

• Simplifying radicals involves expressing the radical in its simplest form by removing any perfect square factors from the radicand (the expression under the radical)
• sqrt(8) = sqrt(4 * 2) = sqrt(4) * sqrt(2) = 2sqrt(2)

Exponents

• Exponents indicate the number of times a base is multiplied by itself
• In the expression x^3, x is the base and 3 is the exponent
• x^3 = x * x * x

Rules 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 (assuming x != 0)
• Negative exponent: x^(-n) = 1 / x^n

Functions

• A function is a relationship between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output
• A function can be represented by an equation, a graph, or a table
• f(x) = 2x + 1 represents a function

Domain and Range

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

Graphing Functions

• Functions can be graphed on a coordinate plane by plotting points and connecting them with a line or curve
• The graph of a function provides a visual representation of the relationship between the input and output values

Linear Functions

• A linear function is a function whose graph is a straight line
• Linear functions can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept

Slope

• The slope of a line is a measure of its steepness and direction
• m = (y2 - y1) / (x2 - x1) gives the slope of a line passing through two points (x1, y1) and (x2, y2)

Y-Intercept

• The y-intercept of a line is the point where the line crosses the y-axis
• It is the value of y when x = 0