Algebra Review

Polynomial Manipulation

Combine like terms, multiply, FOIL, factor, etc.

Rational Expressions

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.

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.

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).

Properties of Exponents and Radicals

10 ax

x

aa

1

yxyx aaa yxy

x

aa

a

xyyx aa xxx baab

x

xx

b

a

b

a

xxx baba

Properties of Exponents and Radicals

21

aa nn aa1

nmmnn m aaa

nnn baab n

n

n

b

a

b

a

baba 22

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.

Systems of Linear Equations

To solve, use either substitution or the addition-subtraction method.

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

Quadratic Formula

02 cbxaxIf

a

acbbx

2

42

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.

Absolute Value Equations

,ax If

ax or ax

Absolute Value Inequalities

Two cases: < or >

axa ax

ax ax or ax

Geometry

Areas

Rectangle: bhA

b

h

Triangle: bhA 21

h

b

Circle:2rA r

Geometry

Perimeters

Rectangle: hbP 22

b

h

Circle: rC 2 r

Geometry

Pythagorean Theorem

x

y

z

222 zyx

Geometry

Boxes

lw

h

Surface Area:

hwlhlwA 222

Volume:

lwhV

Geometry

Cylinders

h

r

Surface Area:

Volume:

hrrA 22 2

hrV 2

Word Problems

Read the problem carefully.

Draw a picture if possible.

Set up a variable or variables, usually for the value you’re asked to find.

Read again and write an equation.

Solve.

Equations of Lines

Standard Form: cbyax

Slope Intercept Form: bmxy

where m is the slope and b is the y-intercept

Slopes of Lines

Given the points (x1, y1) and (x2, y2) on a line, its slope

is 12

12

xx

yym

If the slope is positive the line is increasing:

If the slope is negative the line is decreasing:

Slopes of Lines

Slopes of Lines

If the slope is zero the line is horizontal:

If the slope is undefined the line is vertical:

Slopes of Lines

21 mm

21

1

mm

If two lines are parallel their slopes are equal.

If two lines are perpendicular their slopes are negative reciprocals.

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.

Distance and Midpoint Formulas

The distance d between the points (x1, y1) and (x2, y2) is given by the distance formula:

212

212 yyxxd

The coordinates of the point halfway between the points (x1, y1) and (x2, y2) is given by the midpoint formula:

222121 yy

,xx

• (x1, y1)

• (x2, y2)

•

Conics

Parabolas

Equations of parabolas are quadratic in x or y

2

2

ayx

axy

Vertex at (0, 0)

2

2

kyahx

hxaky

Vertex at (h, k)

Conics

Parabolas

If the x term is quadratic, (ax2), the parabola is vertical.

If a > 0, it opens up

If a < 0, it opens down

Conics

Parabolas

If the y term is quadratic, (ay2), the parabola is horizontal.

If a > 0, it opens right

If a < 0, it opens left

Conics

Circles and Ellipses

Standard form for an ellipse centered at origin:

12

2

2

2

b

y

a

x

-a a

-b

b

Conics

Circles and Ellipses

Standard form for a circle centered at origin with radius r:

222 ryx r-r

r

-r

ConicsHyperbolas

Standard form for hyperbolas centered at origin:

12

2

2

2

b

y

a

x

-a a

12

2

2

2

a

x

b

yb

-b

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.

222 rkyhx

12

2

2

2

b

ky

a

hx

12

2

2

2

a

hx

b

ky 12

2

2

2

b

ky

a

hx

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)

Functions

Notation and evaluating

If f(x) = 3x + 5,

to find f(2), substitute 2 in for x

f(2) = 3(2) + 5 = 11

f(a) = 3(a) + 5

f(joebob) = 3(joebob) + 5

Functions

Notation and evaluating

Note:

hfxfhxf

Functions

Composition of functions

To find f[g(x)], substitute g(x) for x in the f(x) equation

Inverse of a function

The inverse of a function is denoted as xf 1 xf

The inverse of a function f(x) “undoes” what f(x) does.

(this means that and ) xxff 1 xxff 1

Functions

The domain of the range of xf xf 1

The range of the domain of xf xf 1

(this means that the x and y values are reversed on the graphs of a function and its inverse.)

Inverse of a function

Complex Numbers

The imaginary number i is defined as

1i so that 12 i

Complex numbers are in the form a + bi

where a is called the real part and bi is the imaginary part.

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 multiply two complex numbers, use FOIL, taking advantage of the fact that to simplify.12 i

To divide two complex numbers, multiply top and bottom by the complex conjugate of the bottom.

Complex Numbers

Complex solutions to the Quadratic Formula

When using the Quadratic Formula to solve a quadratic equation, you may obtain a result like , which you should rewrite as

4.i2144

In general if a is positive.iaa

Polynomial Roots (zeros)

If f(x) is a polynomial of degree n, then f has precisely n linear factors:

nn cx...cxcxcxaxf 321

where c1, c2, c3,… cn are complex numbers.

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.

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)

Exponentials and Logarithrithms

Logs and exponentials are inverse functions, so if

0 byb x

then xylogb

Exponentials and LogarithrithmsProperties of logarithms

ylogxlogxylog bbb

01 blog

ylogxlogy

xlog bbb

xlogpxlog bp

b

xlogxlogxlogxln e 10

Exponentials and Logarithrithms

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.

To solve an equation involving exponentials, either put

into form , which gives , and then solve for x, or

xgxf bb xgxf

Sequences and Series

Factorial Notation

If n is a positive number, n factorial is defined as

nn!n 1321

with 10 !

For example, 2412344 !

Sequences and Series

An infinite sequence is a list of numbers in a particular order.

The terms of a sequence are denoted as

...,a...,a,a,a n31 2

Sequences and Series

Summation Notation

The sum of the first n terms of a sequence is written as

n

kkn aa...aaa

1321

Sequences and Series

An infinite series is the sum of the numbers in an infinite sequence.

1

31 2k

kn a...a...aaa

Sequences and Series

Arithmetic Sequences

A sequence is arithmetic if the difference between metic if the difference between consecutive terms is constant.consecutive terms is constant.

d...aaaaaa 342312

d is the common difference of the series.

Sequences and Series

Geometric Sequences

A sequence is geometric if the ratio of consecutive terms is if the ratio of consecutive terms is constant.constant.

03

4

2

3

1

2 rr...a

a

a

a

a

a

r is the common ratio of the series.

Sequences and Series

Geometric Series

The sum of the terms in an infinite geometric sequence is called a geometric series.

0

32

k

kn ar...ar...ararara

If , the series has the sum 1r

r

aS

1

Matrices and Determinants

Matrices

An matrix is a rectangular array of numbers with m rows and n columns.

nm

232221

131211

aaa

aaaA is a matrix. 32

Scalar multiplication of a matrix is performed by multiplying each element of a matrix by the same number (scalar).

Matrices and Determinants

Matrices

232221

131211

555

5555

aaa

aaaA

Matrix addition and subtraction is performed by adding or subtracting corresponding elements of the two matrices.

Matrices and Determinants

Matrices

(note: in order to add or subtract two matrices, they must be the same size)

32323131

22222121

12121111

3231

2221

1211

3231

2221

1211

baba

baba

baba

bb

bb

bb

aa

aa

aa

BA

Matrices and Determinants

Determinant of a Square Matrix

The determinant of the matrix is22

dc

baA

dc

baAAdet

dc

ba ad bc