fundamentals of algebra

8/6/2019 Fundamentals of Algebra

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FUNDAMENTALS [email protected]


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Real Numbers

FactoringRational ExpressionsInequalities and absolute value


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REAL NUMBERS

The set of real numbers

Natural numbers N = {1, 2, 3, .. }Whole numbers W = {0, 1, 2, 3, .. }Integers I = { .. -3, -2, -1, 0, 1, 2, 3, .. }Rational numbers

Q = {a/b | a and b integers, b 0}

Irrational numbersReal numbers : all rational and irrationalnumbers


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The set of real numbers

N

W

I

Q

R


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Operations with real numbers

Underaddition and multiplication

a (b + c) = ab + ac distributive

RulesUnder

addition

Under

multiplication

commutative a + b = b + a ab = ba

associative a + (b+c) = (a+b) + c a (bc) = (ab)c

identity a + 0 = a a. 1 = 1 . a

inverse a + (-a) = 0 a(1/a) = 1


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Polynomial in x is an expression of the form

anxn + an-1xn-1 + .. + a1x + a0

Where n is a nonnegative integer and

a0, a1, , an are real numbers, with

an 0

Adding and subtracting polynomialsMultiplying polynomials

Polynomial in two variablerr


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EXERCISE

Compute :

a.(2x + y)2b. (3a 4b)2c.(1/2 x 1) (1/2 x + 1)


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FACTORING

The process of expressing polynomial as a

product of two or more polynomials.

Formula

a2 b2 = (a + b)(a b)a2 + 2ab + b2 = (a + b)2

a2 2ab + b2 = (a b)2

a3

+ b3

= (a + b)(a2

ab + b2

)a3 b3 = (a b)(a2 + ab + b2)


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EXAMPLE

x2 36

8x2 2y2

9 a6

x2 + 8x + 16

4x2 4xy + y2

x3 + 27

8x3 y6


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EXERCISE

1.x2 4

2.x2 + 3x 4

3.u4 v4


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RATIONAL EXPRESSIONS

Simplifying Rational Expressions

Example :

Exercise

341

343 xx

b

a

bc

ac

!

!

16

)1()4(

12

443

34

32

2

2

2

2

2

k

kk

x

xx

xx

xx


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Operation

ExerciseSimplify

R

QP

R

Q

R

PnSubtractio

R

QP

R

Q

R

PAddition

R

S

Q

P

S

R

Q

PDivision

QSPR

SR

QPtionMultiplica

!

!

!z

!

:

:

.:

.:

xhx

y

y

x

x

11

3

24

4

43


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Inequalities

Symbols inequalities : < , > , ,

Strict inequalities, example : 3 < 5.

Conditional inequalities, example :x> 100.

Double inequalities, example : 0


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Interval

Finite Intervals

Open interval, (a, b) = {x| a


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Solving Inequalities

Property

If a < b and b < c, then a < c

If a < b, than a + c < b + c

If a < b and c > 0. then ac < bc

If a < b and c < 0, then ac > bc

Solution set : the set of all real numbers

satisfying the inequalities.

Example : solve 3x 2 < 7 and -1 e 2x - 5 < 7


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Exercise

Solve :

3x 2 < 4x + 8

7x 1 e 10x + 4

-4x + 10 u -10 + x

-6 < 2x + 3 < -1

12 u x + 16 u -20

-4x + 10 e x e 2x + 6

10 + x e 2x 5 e 25


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Solving Inequalitiesby Factoring

Steps :

Set the polynomial in the inequality equal to 0

Factor the polynomial

Construct a sign diagram for the factors of the

polynomialDetermine the intervals that satisfy the given

inequality

Example :

x2 x < 6 x 2 2x 8 < 0

x2 2x - 15 > 0 3x 2 x + 8 > 10


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Exercise

x2 16 e 0

x2 9 u 0

x2 + 2x 8 u 0

2x2 + 5x + 3 < 0

6x2 + x 12 > 0


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AbsoluteValue

The absolute value of a number x is denoted by

|x| and is defined by :

|x| =x if xu 0

|x| = -x if x< 0

Solving the absolute value :

|x| < a -

a a x< -a atau x> a


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Example

|x| < 1

|x| > -4

|x 5| < 10

|2x + 3| > 5


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Exercise

|x| e 4

|x| > 2

|x 4| < 1,5

|3x 5| u 1

|4 2x| < 2

fundamentals of algebra

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