Intermediate Algebra
The set of real numbers
Natural numbers
Whole numbers
Integers
Rational numbers
Irrational numbers
Real numbers
Operations on real numbers
Properties of real numbers
Decimal Numbers
Decimal representation of rational numbers or fractions
Expanded form of decimal number, decimal fractions
Terminating decimals

Terminating decimals to fraction
Purely recurring decimals to fraction

Exponential or scientific notation of decimal numbers
Order of operations
Properties of exponents
The rules for powers or exponents
Simplifying an exponential expression
x2 = aa > 0
Square root
Properties of square roots
Adding, subtracting, multiplying and dividing square roots
Rationalizing a denominator
The graph of the quadratic function  f (x) = x2
Translation of the source quadratic function in the direction of the y-axis,
quadratic function of the form  f (x) = x2 + y0
x2 = aa > 0
the inverse of the square of x (or quadratic) function
Definition of the inverse function
The graph of the principal square root function
Translation of the principal square root function in the direction of the x-axis
Radicals (Roots) and/or Exponentiation with Fractional (Rational) Exponent

Rules and properties of radicals and/or fractional exponents

Rationalizing a denominator

Algebraic expressions - preliminaries
Evaluating algebraic expressions
Simplifying algebraic expressions
Expanding algebraic expression by removing parentheses (i.e. brackets)
The square of a binomial - the perfect square trinomial
The square of a trinomial
The cube of binomial
The binomial expansion algorithm
The difference of two squares, multiplying
The difference of two squares, factoring
The difference of two cubes
The sum of two cubes
Factoring and expanding algebraic expressions, rules for transforming algebraic expressions
Factoring algebraic expressions - methods, the greatest common factor
Factoring by grouping
Perfect square trinomials - the square of a binomial
The difference of two squares
The sum and difference of cubes
Factoring polynomials
Using a variety of methods including combinations of the above to factorize algebraic expressions

Solving linear equations
Word problems that lead to simple linear equations
Number problems

Mixture problems
Work problems
Time and travel (distance) problems
Geometry word problems
Miscellaneous word problems
Variable expressions and formula problems
Solving a formula for a specified variable, transposition of a formula (changing the subject of a formula)
Absolute value equations
Solving absolute value equations

Solving inequalities
Properties of inequalities
Examples of solving single linear inequalities
Solving compound (double) inequalities

Solving linear inequalities with absolute value

Definition of a polynomial or polynomial function
Source or original polynomial function
Translating (parallel shifting) of the polynomial function
Coordinates of translations and their role in the polynomial expression
Transformations of the polynomial function applied to the quadratic and cubic functions
Zeros of a polynomial function

Multiplication of polynomials
Division of polynomials
Factoring polynomials and solving polynomial equations by factoring
Graphing polynomial functions
Zero polynomial
Constant function
Linear function
Transformations of the graph of the quadratic function
Cubic functions
Graphs of cubic functions

Linear function  f (x) = mx + c

Lines parallel to the axes, horizontal and vertical lines
The point-slope form of a line
Parallel and perpendicular lines

The system has, a single solution, no solution or has infinitely many solutions

Independent equations, inconsistent equations and dependent equations

Method of expanding a determinant of a rank n to cofactors

Solving and graphing systems of linear inequalities
Solving and graphing systems of linear inequalities in two variables

Method of solving rational inequalities
The graph of the translated equilateral (or rectangular) hyperbola
Equations with Rational Expressions
Solving rational equations
Rational equations - Linear equations

basic definitions
Imaginary unit
Complex numbers
Real and imaginary parts
The set of all complex numbers C
Absolute value, modulus of a complex number
Addition and subtraction of complex numbers
Multiplication and division of complex numbers
Polar or trigonometric notation of complex numbers
Multiplication and division of complex numbers in the polar form
Exponentiation and root extraction of complex numbers in the polar form
Powers and roots of complex numbers, use of de Moivre’s formulas
Euler’s formula, relationship between trigonometric functions and the complex exponential function

Simple interest
Amount of interest, principal (initial amount), interest rate and amount after n years
Compound interest
Periodic compounding
Continuous compounding
Exponential growth and decay, application of the natural exponential function

Vieta’s formula
Quadratic function or the second-degree polynomial

Roots or zeros of the function, axis of symmetry and y-intercept
Transformations of the graph of the quadratic function

Function definition, notation and terminology
Domain, range and codomain
Evaluating a function
Composition of functions (a function of a function)

Types of functions - basic classification
Algebraic functions and Transcendental functions
Algebraic functions
The polynomial function
Rational functions
Reciprocal function
Transcendental functions
Exponential and logarithmic functions, inverse functions
Trigonometric (cyclometric) functions and inverse trigonometric functions (arc-functions)
The graph of a function
Properties and characteristic points of the graph
Domain and range
Roots or zero function values, x-intercepts, y-intercepts
Increasing/decreasing intervals
Continuity and discontinuity
Vertical, horizontal and oblique or slant asymptotes
Stationary points and/or critical points
Turning points (extremes, local or relative maximums or minimums)
Inflection points and intervals of concavity

Transformations of original or source function
How some changes of a function notation affect the graph of the function

Reflections of the graph of a function
The graphs of the elementary functions

algebraic functions

The source or original polynomial function
Translating (parallel shifting) of the polynomial function
Coordinates of translations and their role in the polynomial expression

The graph of the quadratic function
The graphs of the cubic function
The graphs of the quartic function
Translated power function
The graphs of rational functions

reciprocal function, equilateral or rectangular hyperbola
Translation of the reciprocal function, linear rational function
The graphs of the algebraic and transcendental functions
Transcendental functions - The graphs of transcendental functions
The graph of the exponential function
The graph of the logarithmic function
Trigonometric (cyclometric) functions and inverse trigonometric functions (arc functions)
The graphs of the trigonometric functions and inverse trigonometric functions or arc-functions
The graph of the sine function
The graph of the cosine function
The graph of the arc-sine function and the arc-cosine function
The graph of the tangent function and the cotangent function
The graph of the arc-tangent function and the arc-cotangent function
The graph of the cosecant function
The graph of the secant function
arc-cosecant and the arc-secant function

Inverse functions

Logarithmic functions

Rules and properties of logarithms
Natural logarithm, common logarithm

Exponential equations

 Conic Sections
Circle
General equation of a circle with the center S (p, q) - translated circle
Equation of the circle with the center at the origin O(0, 0)
Circle through three points

Line circle intersection
Equation of a tangent at a point of a circle with the center at the origin

O(0, 0)

Angle between a line and a circle
Mutual position of two circles

Definition and construction

Constructions of an ellipse

Major axis, minor axis, and vertices

The parametric equations of the ellipse

Ellipse and Line
Intersection of ellipse and line - tangency condition
Equation of the tangent at a point on the ellipse

Ellipse and line examples

Equation of the hyperbola

Equilateral or rectangular hyperbola

Equation of the hyperbola in vertex form

Examples of hyperbola

Translation of equilateral or rectangular hyperbola with the coordinate axes as its asymptote

Polar and pole of the hyperbola

The area of a triangle which the tangent at a point on the hyperbola forms with asymptotes
The tangency point bisects the line segment of the tangent between asymptotes
The parallels to the asymptotes through the tangency point intersect asymptotes
The equation of the equilateral or rectangular hyperbola with the coordinate axes as its asymptotes

of the parabola

Common points of a line and a parabola

Properties of the parabola

Construction of the tangent at the point on the parabola
Construction of the tangents from a point exterior to the parabola
Parabola and line, examples
Conics, a Family of Similarly Shaped Curves – Properties of Conics
Conics, a Family of Similarly Shaped Curves - Properties of Conics
Dandelin's Spheres - proof of conic sections focal properties
Proof that conic section curve is the ellipse

 Sequences and Series

Arithmetic sequence/progression

The sum of the first n terms of an arithmetic sequence
The sum of the first n natural numbers
Arithmetic sequences, examples
Geometric sequence/progression

The sum of the first n terms of a finite geometric sequence, geometric series
Geometric sequences, examples
Recursive definition and the recursion formula

Finite and infinite, convergent and divergent series

The sum of an infinite geometric sequence, infinite geometric series
The sum of an infinite converging geometric series, examples

Mathematical induction examples

Pascal's triangle
The binomial theorem, sigma notation and binomial expansion algorithm
The binomial theorem and binomial expansion algorithm examples
 Combinatorics - Combinatorial Analysis
Permutations, Combinations and Variations

Permutations of n objects some of which are the same

Variations or permuted combinations (permutations without repetition)
Variations with repetition (or permuted combinations with repetition)