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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