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INTRODUCTORY ALGEBRA • A-571-11

TERMINOLOGY

Rational numbers Ratios of integers: 3/4, –8/5, 127/16,

–3/1 = –3

Irrational numbers Numbers that cannot

be expressed as ratios

of integers:

Algebraic expressions Combinations of

numbers and letters:

Terms of an Products or quotients of numbers and letters:

algebraic expression

is an algebraic expression, not a term

Monomial Algebraic expressions consisting of one term

Binomial Algebraic expressions consisting of two terms

Multinomial Algebraic expressions

consisting of more

than two terms:

Polynomial Algebraic expressions consisting of several

terms that are integral and rational in the

lettered terms: 5x4– 6x3+ x2– 2x + 1

but 3/x + y, are not polynomial

Degree of monomial Sum of exponents of lettered symbols:

4x3y2z5is of degree 10

Degree of polynomial Exponent of the term of the highest degree:

5x4– 6x3+x2– x + 1 has a degree of 4

INTEREST

P= principal

I= accumulated interest

A= principal + interest

R= interest rate per year

N= number of years

I= P ¥R ¥N

A= P(1 + RN)

Example

• Interest to be paid on

$4,000 borrowed at 10%

for 5 years

I = PRN = 4000 ¥0.10 ¥5

= $2,000

• Principal to be invested at

7% that will yield $10,000

in 5 years

71

323

,,,

/

−

π

e

56 2 5

26

2

3

ab xy

bc d

a

−+

−

,,

51

251

2

2323

xy ab

cxy ab

c

, but −

63

22

x

y

z

xxyz+ +

43z+

a, b, c, u, v, w

= real constants

A, B, C, D, E

= algebraic expressions

f(x), g(x)

= functions of x

SIMPLE INTEREST

PA

RN

=+=+×

=

1

10 000

1007 5

740741

,

.

$, .

P= principal

I= compound interest

A= compound amount

R= interest rate per period

N= number of interest periods

I= A – P

A= P(1 + i)N

Example

• Compound amount Areceived

and compound interest Iearned

on principal of $5,000

compounded semi-annually at 5%

over 2 years

i= 0.05/2 = 0.025

N= 4

P= $5,000

A= P(1 + i)N= 5000 (1 + 0.025)4

= $5,519.06

I= A – P = 5519.06 – 5000

= $519.06

COMPOUND INTEREST

QUADRAT I C EQUAT I O N S

• Every quadratic equation (ax2+ bx + c = 0)in one unknown

has exactly two roots (x1and x2) that can be real, equal, or

complex

Notes

• If one is complex, then the 2nd is its conjugate: A ± Bi

• The sum of the roots is given by: x1+ x2= –b/a

• The product of the roots is given by: x1x2= c/a

QUADRATIC FORMULA THE ROOTS

• The roots of a

quadratic equation are

given directly by the

relation:

xbb ac

a

12

24

2

,=−± −

Example

Given:

Substitute in:

and

3850

864453

23

153

2

12

12

xx

x

xx

++=

=−± − ××

×

=− =−

,

/∴

Nature of the Roots

The Discriminant, D = b2– 4ac

D = b2– 4ac > 0 Roots are

real and

unequal

D = b2– 4ac = 0 Roots are

real and

equal

D = b2– 4ac < 0 Roots are

complex

conjugates

D = – 4ac < 0 Roots are

pure

imaginary

Introductory Algebra

Introductory Algebra

TM

permacharts

DEALING WI T H FRACTIONS

RULES EXAMPLE

Rule 1

The numerator/denominator

can be multiplied/divided by

the same factor

Rule 2

To add/subtract fractions,

write each fraction in terms

of a common denominator,

preferably the lowest

common denominator (LCD)

Rule 3

To multiply fractions, multiply

the numerators and

denominators separately

Rule 4

To divide by a fraction, invert

the fraction and multiply

Rule 5

To divide sums or differences

of fractions, reduce the

numerator and denominator

to single fractions using LCDs,

and then divide the result

x

y

x

y

x

y

xz

yz

=−

−==

3

3

/

/

72

1

31

71231

1

943

1

2

2

2

2

2

2

xx xLCD x x

xx x x

xx

xx

xx

−−==−

=−+ + −

−

=−−

−

()

() ()

()

()

2

3

612

3

4

22

a

b

b

a

ab

ab a

×= =

aab

a

b

a

aa ab

ba

b

b

+÷+=+

+

=+

+=

2

3

21

6

62

213

21 2

21 2

22

()

()

()

a

aa

a

aa

aa

aa

aa

aa

aa aa

a

+

−−

+

+−

+−−

−+

−+ +

+−

−

+=

=++−+−

=

1

11

1

1111

11

11

11

11

22

22

21 21

22

()()

()()

()()

()()