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