Algebra Review

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Algebra Review
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Polynomial Manipulation
Combine like terms, multiply, FOIL, factor, etc.
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Rational Expressions
To multiply rational expressions (fractions),
multiply the numerators, and multiply the
denominators.
To divide rational expressions, invert the second
expression and then follow the rules for
multiplication.
If possible, it may be helpful to factor the
numerators and denominators before multiplying.
To add or subtract rational expressions, find the
least common denominator, rewrite all terms with
the LCD as the new denominator, then combine like
terms.
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Rational Equations
To solve rational equations, multiply both sides
of the equation by the LCD of both sides of the
equation, and then solve.
Be sure to check your answers by substituting
them back into the original equation, in case
your solution causes the original expression to
become undefined (a zero in the denominator).
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Properties of Exponents and Radicals
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Properties of Exponents and Radicals
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Equations Involving Radicals
If only one square root is present, isolate it on
one side of the equal sign, square both sides and
solve.
If two square roots are present, put one on each
side of the equal sign, square both sides and
solve.
When solving equations containing radicals,
extraneous solutions are often introduced, which
means you must check your answer in the original
equation.
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Systems of Linear Equations
To solve, use either substitution or the
addition-subtraction method.
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Quadratic Equations
First put zero on one side of the equal sign and
everything else on the other side.
Then use reverse FOIL ( )(
) 0
or, use the Quadratic Formula
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Quadratic Formula
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Inequalities
Use the same methods as in solving equations,
with the exception that if you multiply or divide
both sides of the inequality by a negative
number, it reverses the order of the inequality.
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Absolute Value Equations
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Absolute Value Inequalities
Two cases lt or gt
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Geometry
Areas
Rectangle
Triangle
Circle
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Geometry
Perimeters
Rectangle
Circle
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Geometry
Pythagorean Theorem
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Geometry
Boxes
Surface Area
Volume
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Geometry
Cylinders
Surface Area
Volume
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Word Problems
Read the problem carefully.
Draw a picture if possible.
Set up a variable or variables, usually for the
value youre asked to find.
Read again and write an equation.
Solve.
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Equations of Lines
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Slopes of Lines
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Slopes of Lines
If the slope is positive the line is increasing
If the slope is negative the line is decreasing
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Slopes of Lines
If the slope is zero the line is horizontal
If the slope is undefined the line is vertical
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Slopes of Lines
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Graphing Lines
The x-intercept is the point where the line
crosses the x-axis, found by setting y 0 and
solving for x.
The y-intercept is the point where the line
crosses the y-axis, found by setting x 0.
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Distance and Midpoint Formulas
The distance d between the points (x1, y1) and
(x2, y2) is given by the distance formula

The coordinates of the point halfway between the
points (x1, y1) and (x2, y2) is given by the
midpoint formula
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Conics
Parabolas
Equations of parabolas are quadratic in x or y
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Conics
Parabolas
If the x term is quadratic, (ax2), the parabola
is vertical.
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Conics
Parabolas
If the y term is quadratic, (ay2), the parabola
is horizontal.
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Conics
Circles and Ellipses
Standard form for an ellipse centered at origin
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Conics
Circles and Ellipses
Standard form for a circle centered at origin
with radius r
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Conics
Hyperbolas
Standard form for hyperbolas centered at origin
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Conics
If the curve is centered at (h,k), replace the x
in the equation with x-h and replace the y with
y-k.
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Functions
Domain and range
The domain of a function f(x) is the set of all
possible x values. (the input values)
The range of a function f(x) is the set of all
possible f(x) values. (the output values)
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Functions
Notation and evaluating
f(2) 3(2) 5 11
f(a) 3(a) 5
f(joebob) 3(joebob) 5
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Functions
Notation and evaluating
Note
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Functions
Composition of functions
To find fg(x), substitute g(x) for x in the
f(x) equation
Inverse of a function
The inverse of a function f(x) undoes what
f(x) does.
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Functions
Inverse of a function
(this means that the x and y values are reversed
on the graphs of a function and its inverse.)
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Complex Numbers
Complex numbers are in the form a bi where a is
called the real part and bi is the imaginary part.
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Complex Numbers
If a bi is a complex number, its complex
conjugate is a bi.
To add or subtract complex numbers, add or
subtract the real parts and add or subtract the
imaginary parts.
To divide two complex numbers, multiply top and
bottom by the complex conjugate of the bottom.
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Complex Numbers
Complex solutions to the Quadratic Formula
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Polynomial Roots (zeros)
If f(x) is a polynomial of degree n, then f has
precisely n linear factors
This means that c1, c2, c3, cn are all roots of
f(x), so that f(c1) f(c2) f(c3) f(cn)
0
Note some of these roots may be repeated.
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Polynomial Roots (zeros)
For polynomial equations with real coefficients,
any complex roots will occur in conjugate pairs.
(If a bi is a root, then a - bi is also a root)
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Exponentials and Logarithms
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Exponentials and Logarithms
Properties of logarithms
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Exponentials and Logarithms
Equations
To solve a log equation, rewrite it as an
exponential equation, then solve.
take the log of both sides and use the properties
of logs to simplify, then solve.
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Sequences and Series
Factorial Notation
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Sequences and Series
An infinite sequence is a list of numbers in a
particular order.
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Sequences and Series
Summation Notation
The sum of the first n terms of a sequence is
written as
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Sequences and Series
An infinite series is the sum of the numbers in
an infinite sequence.
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Sequences and Series
Arithmetic Sequences
A sequence is arithmetic if the difference
between consecutive terms is constant.
d is the common difference of the series.
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Sequences and Series
Geometric Sequences
A sequence is geometric if the ratio of
consecutive terms is constant.
r is the common ratio of the series.
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Sequences and Series
Geometric Series
The sum of the terms in an infinite geometric
sequence is called a geometric series.
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Matrices and Determinants
Matrices
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Matrices and Determinants
Matrices
Scalar multiplication of a matrix is performed by
multiplying each element of a matrix by the same
number (scalar).
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Matrices and Determinants
Matrices
Matrix addition and subtraction is performed by
adding or subtracting corresponding elements of
the two matrices.
(note in order to add or subtract two matrices,
they must be the same size)
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Matrices and Determinants
Determinant of a Square Matrix