ALGEBRA

l e a r n • r e f e r e n c e • r e v i e w

College Mathematics

College Mathematics

ALGEBRAIC LAWS

SPECIAL PRODUCT FORMULAS

LOGARITHMS

Sign Laws

Commutative Laws Associative Laws

Distributive Laws Laws Involving Zero

Exponent Laws

• Given an = b, then the following are true:

Logarithm: n = logab = logarithm of bto the base a

Anti-logarithm: b= antilogan= anti-logarithm of nto the base a

Common logarithm (log): Logarithm to the base 10

Natural logarithm (In): Logarithm to the base e= 2.718282

Remember: Logarithms are exponents

Laws of Logarithms

Change of Base

Special Relations

SOLUTION OF EQUATIONS

Two Linear Equations in Two Variables

Algebraic Procedure

Solve either equation for one unknown in terms of the other

Substitute solution into second equation and solve to obtain

first unknown

Substitute this unknown into either of the original equations

and solve for second unknown

Graphical Procedure

Equations: x+ y= 1 and x+ y= 0

Solution: None

Reason: • Lines do not intersect

• Equations are inconsistent,

as x+ycannot be both

0 and 1

Equations: x+ y= 1 and x– y= 0

Solution: One

Reason: • Lines intersect at single

point: x= 1/2 and y= 1/2

• Unique solution

Equations: x+y= 1 and 2x+ 2y= 2

Solution: Infinite number of solutions

Reason: • Lines are identical

• Equations are dependent

(i.e., multiples of each other)

Quadratic Equations in One Variable

• The roots of the quadratic equation ax2+ bx + c= 0

are given by the relation:

• The nature of the roots is determined by the discriminant

D = b2– 4ac as follows:

−−

()

=

()

=−

()

−

()

=

()

=−

()

−

()

=

()()

=

−

()

=−

()

=− −

()()

=−

()

=− −

()()

=−

()()

=−

AAA ABABAB ABABAB

AAA ABABAB AB ABAB

AB BA AB BA+=+ =

AA A A A AB AB

AA A A A AB AB

AAAAB BA

AA

mnmn nmmm

mn mn m m mm m

mnmn mm

xyx

y

n

== =

()

== =

()

()

==

()

=

()

=

+

−−

−

1

1

1

0

ABC ABCABC ABC+ +

()

=+

()

+

()

=

()

AB A ABB AB A AB AB B

AB ABAB AB ABAABB

ABC A B C ABC BC

AB

±

()

=± + ±

()

=± + ±

−=+

()

−

()

±=±

()

+

()

+++

()

=+++

()

+++

()

++

()

±

(

222 332 23

22 33 2 2

222 2

233

22

))

=± + −

()

±−

()

−

()

+

−− −

nnn n n

AnABnn AB nn n AB

1

1

12

12

12 3

122 33

()( ) ()( )( )

AB A B

A

A

===

=∞

=

000

0

0000

means and or

or does not exist

is indeterminate

AB C AB AC

AB A B

mmm

+

()

=+

()

=

log log log log log

log log log log log

aaaa

na

aaaa

na

xy x y x n x

xy x y x nx

•

()

=+ =±

()

=− =

±

1

log log log log . ln ln . log

aAA

xxax xx x===0 43429 2 30259

eex x

xx x xlnlog

ln log====1010

xABi

DxBi

=

=±

==±

2Roots are real and unequal

Roots are real and equal

Roots are complex conjugates:

Roots are pure imaginary:

1,2

1,2

xbb ac

a

12

24

2

=−± −

,

Binomial Series

COMPLEX NUMBERS

Imaginary unit

Complex number

Conjugate

complex numbers

Modulus r:

Argument

q

:

Complex number

(polar form)

ii nin=−=− −=1 1

2

or Always write:

Given by and : ar br= =cos sin

θθ

0

zabi z

a

" "

"

Example:

If

23i

then z is pure imaginary ; 0b"If then z is real

zabi z abi

12

=+ =−and

Modulus of z is z = rb" ()

/2212

a

zr i= +(cos sin )

θθ

Equality

Addition/subtraction

Multiplication

Multiplication of conjugates

Multiplication in polar form

Division

Division in polar form

Integer powers of i

Algebraic Operations

abicdi ac bd"""means and

()()()()a bi c di a c b d i+±+ = ±+±

()()( )( )a bi c di ac bd ad bc i++= − + +

()()a bi a bi a b+−=+

22

zz rr i

12 12 1 2 1 2

•=+++[cos( ) sin( )]

θθ θθ

abi

cdi

abicdi

cdicdi

ac bd bc ad i

cd

+

+=+−

+−

=++−

+

()()

()()

()()

22

zz rr i

12 12 1 2 1 2

=−+−( )[cos( ) sin( )]

θθ θθ

iiiiiii

2341 2

11 1=−=−= =− =−

−−

;;; ;

© 1996-2014 Mindsource Technologies Inc.

COLLEGE MATHEMATICS • A-632-71

TM

permacharts

www.permacharts.com

Y

X

1

1

Y

X

1

1

Y

X

1

1

Db ac

Db a

=-4>0 Rootsarerealandunequal

=-4

2

2cc

Db ac

=0 Rootsarerealandequal

=-4<0 Roots

2 are complex conjugates : = ±

=-4 <0

1

Dac( ) Rootsarepureimaginary =±

1

b=0 x Bi:

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