ALGEBRA - Solved problems contents
     Sets
     Relations
     Binary numbers
         Binary operations
         The method of subtraction by adding complement of the subtrahend applied to decimals
         The method of subtraction by adding complement of the subtrahend applied to binary numbers
     Natural numbers and integers, rules and properties
     Operations on integers
         Addition and subtraction
         Multiplication and division
         Order of operations - parenthesis as grouping symbols
     The integer prime-factorization
        The greatest common divisor (GCD)
        The least common multiple (LCM)
     Operations on real numbers
     Decimal representation of rational numbers or fractions
        Terminating decimals
         Recurring decimals
         Purely recurring decimals
         Mixed recurring decimals
     Order of operations
     Fractions
     Proper and improper fractions, mixed numbers
         Converting and reducing fractions
         Comparing fractions
         Decimal fractions
     Adding and subtracting like fractions
     Adding and subtracting fractions
     Multiplying fractions
        Dividing fractions, reciprocal fractions
         Simplifying complex fractions
     Adding and subtracting decimals
     Multiplying and dividing decimals by powers of ten or by decimal units
     Multiplication and division of decimal numbers
     Rational numbers
         Equivalent rational numbers or fractions
     Reducing or simplifying rational numbers or fractions
     Adding and subtracting rational numbers or fractions
     Multiplication of rational numbers
         Division of rational numbers, reciprocal fractions (multiplicative inverse)
         Simplifying complex or compound fractions
     Properties of real numbers
     Exponentiation – powers or indices
     Adding and subtracting same powers
     The rules for powers (or exponents)
         Simplifying an exponential expression
         Scientific notation
     Quadratic equation  x2 = aa > 0
     Square root
     Properties of square roots
        Rationalizing a denominator
     Radicals (roots) and/or exponentiation with fractional (rational) exponent
        Rules and properties of radicals
     Rules and properties of radicals and/or fractional exponents
        Simplifying radical expressions
     Operations on radical expressions
        Adding, subtracting and multiplying radicals
        Rationalizing a denominator
        Radicals and/or fractional (rational) exponents
     Algebraic expressions
        Simplifying algebraic expressions
        Evaluating algebraic expressions
     Expanding algebraic expression by removing parentheses ( brackets)
        The square of a binomial (or binomial square)
        Squaring trinomial (or trinomial square)
        Cube of a binomial
     Factoring algebraic expressions
        Factoring algebraic expression by finding (determining) a common factor
        Grouping like terms, grouping and factorizing four terms
        The square of a binomial - perfect squares trinomials
        Difference of two squares
     Factoring quadratic trinomials
     Factoring cubic or a third degree polynomial
        Sum and difference of cubes
        Using a variety of methods including combinations of the above to factorize expressions
     Factoring and expanding algebraic expressions, rules for transforming algebraic expressions
     The binomial expansion algorithm - the binomial theorem
     Rational expressions
     Simplification of rational expressions, reducing to lowest terms
     Addition and subtraction of rational expressions
     Multiplication and division of rational expressions
     Solving complex rational expressions
     Ratios and proportions
        Ratios and proportion properties
     Direct and inverse proportion
     Percent or percentage
     Percent, decimal number and fraction conversions
     Basic percent formulas
     Percent increase or decrease
     Interest calculations
     Simple interest
     Compound interest
        Periodic compounding
        Continuous compounding
     Exponential growth and decay, application of the natural exponential function
     Linear equations in one variable
     Solving linear equations
        Equations with rational expressions
     Word problems that lead to simple linear equations
     Number problems
     Age problems
     Mixture problems
     Work problems
     Time and travel problems - Distance, rate (or speed) and time relations
     Geometry word problems
     Miscellaneous word problems
     Variable expressions (expressions involving variables) 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
     Linear equation with absolute value, graphic solution
     Linear inequalities
     Solving linear inequalities
        Solving compound (double) inequalities
     Systems of linear inequalities
     Solving and graphing systems of linear inequalities in two variables
     Rational Inequalities
     Absolute value inequalities
     Graphing linear equation, linear function (first degree polynomial)
     The linear function  f(x) = mx + c
     Equations of straight line
        The point-slope form of a line
        The two point form of the equation of a line
     Absolute value functions
     The graph of the absolute value function  f(x) = | x |
        The graph of the absolute value of a linear function  f(x) = | ax+ b |
     Simultaneous linear equations
     System of two linear equations in two unknowns (variables)
        Solving systems of equations - substitution and comparison method
        Addition or elimination method
     System of linear equations word problems
     Solving systems of equations graphically
     Cramer’s rule (using the determinant) to solve systems of linear equations
        Solving system of two equations in two unknowns using Cramer's rule
        Solving system of three equations in three unknowns using Cramer's rule
     Method of expanding a determinant of a rank n to cofactors
     Imaginary and complex numbers
        Imaginary numbers
     Complex numbers
        Addition and subtraction of complex numbers
        Multiplication and division of complex numbers
        Polar or trigonometric notation of complex numbers
     Exponentiation and root extraction of complex numbers in the polar form - de Moivre's formula
     Polynomial and/or polynomial functions and equations
        Addition and subtraction of polynomials
        Multiplication of polynomials
        Division of polynomials
     Quadratic equation
        Solving quadratic equations by completing the square, the quadratic formula
        Quadratic equation word problems
        Solving quadratic equations by factoring, Vieta’s formula
        Factoring polynomials and solving polynomial equations by factoring
     Polynomial functions
        Transformations of the polynomial function applied to the quadratic and cubic functions
     Quadratic function and equation
        Quadratic inequalities
        Quadratic equations with absolute value
     Cubic functions or the third degree polynomial
     Equations with rational expressions
     Rational equations - Linear equations
     Rational equations - Quadratic equations
     Binomial equations
        Solving binomial equations
     Equations reducible to quadratic form - biquadratic equations
        Solving biquadratic equations or equations reducible to quadratic
     Radical or irrational equations
        Solving irrational or radical equations
     Rectangular (two-dimensional, Cartesian) coordinate system
     Midpoint of a line segment
     The distance formula
     Dividing a line segment in a given ratio
     The area of a triangle
     Lines parallel to the axes, horizontal and vertical lines
     Polar coordinate system
     Polar and Cartesian coordinates relations
        Equation of a line in polar form
        Lines parallel to the axes, horizontal and vertical lines
        Polar equation of a line
        Equation of a circle in polar form
     Parametric equations
        The parametric equations of a line
        The parametric equations of a quadratic polynomial, parabola
        The parametric equations of the parabola, whose axis of symmetry is parallel to the y-axis
        The parametric equations of the parabola, whose axis of symmetry is parallel to the x-axis
     Use of parametric equations, example
        Intersection point of a line and a plane in three dimensional space
     Introduction to functions
     Evaluating a function
     Composition of functions (a function of a function)
     Inverse function
     The graph of a function
        Symmetry of a function, parity - odd and even functions
     Transformations of original or source function
        How some changes of a function notation affect the graph of the function
        Translations of the graph of a function
        Reflections of the graph of a function
     Cubic functions, the polynomial of the third degree
        Source and translated cubic function
        Graphing cubic functions, examples
     The polynomial of the first type (the power or monomial function)
        The polynomial of the first type example, the graph of the sextic
     Exponential and logarithmic functions and equations
        Inverse functions
     The graph of the exponential function
     The graph of the logarithmic function
     Translated logarithmic and exponential functions
     Rules and properties of logarithms
     Exponential and logarithmic equations
     Exponential equations
     Logarithmic equations
     Rational functions
     The graph of the reciprocal function, equilateral or rectangular hyperbola
     Translation of the reciprocal function, linear rational function
     Graphing rational functions
        The oblique or slant asymptote of rational functions example
     Trigonometric functions and equations
     The graphs of the sine and cosine functions
        The function  y = asin x
        The function  y = sin bx
        The function  y = sin (x + c)
     The function  y = asin (bx + c)
     Trigonometric equations
        The equation  sin x = a
        The equation  cos x = a
        The equation  tan x = a
        The equation  cot x = a
     The equation  sin (bx + c) = m
        The equation  cos (bx + c) = m
        The equation  tan (bx + c) = m
        The equation  cot (bx + c) = m
     Equations that can be written as  f · g = 0
     Trigonometric equations of quadratic form
     Equations of the type  a · cos x + b · sin x = c
        Introducing an auxiliary angle method
        Introducing new unknown  t = tan x/2
     Homogeneous equations in sin x and cos x
        Homogeneous equations of first degree  a sin x + b cos x = 0
        Homogeneous equations of second degree  a sin2 x + b sin x · cos x + c cos2 x = 0
     The basic strategy for solving trigonometric equations
        Solving trigonometric equations examples
     Sequences and series
     Sequences
        Arithmetic sequence or progression
        Geometric sequence or progression
     Recursive definition and the recursion formula
     The sum of an infinite geometric sequence, infinite geometric series
     Converting recurring decimals (infinite decimals) to fraction
        Converting purely recurring decimals to fraction
        Converting mixed recurring decimals to fraction
     Mathematical induction
        Mathematical induction examples
     The binomial theorem
        Binomial coefficients and Pascal's triangle
     The binomial theorem, sigma notation and binomial expansion algorithm
     Matrices and determinants
     Matrices
        Vectors
        Transposition
     Matrix addition and subtraction
     Scalar multiplication of a matrix
     Matrix multiplication
        The matrix-vector product
     The determinant of a matrix
        Expanding a determinant by cofactors
        Calculating the value of the determinant of a 3 ´ 3 matrix
        Properties of determinants
     The inverse of a matrix
     Solving systems of linear equations using matrices
     Combinatorics or combinatorial analysis
        Permutations, combinations and variations
     Permutations
        Permutations of n objects some of which are the same
     Combinations
        Combinations with repetition
     Variations or permuted combinations (permutations without repetition)
        Variations with repetition (or permuted combinations with repetition)
     Probability
     The probability of mutually exclusive events
     The probability of not mutually exclusive events
     The probability of two independent events
     The probability of two dependent events
     Sequences and limits
     Infinite sequences
        Graphing the terms of a sequence on the number line
        The limit of a sequence
        Convergence of a sequence
        Verifying the convergence of a sequence from the definition
        The Cauchy criterion or general principle of convergence
     Operations with limits
     Series
     Infinite series
        The sequence of partial sums
        The sum of the series
        Convergence of infinite series
        Divergence of infinite series
     Harmonic series
     Geometric series
     P-series
     Alternating series
        Absolute convergence
        Conditional convergence
     Series of positive terms
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