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Transcript of Fundamental Concepts of Algebra 1 1.1 Real Numbers Objective: Students will be introduced to the...
Fundamental Concepts of Algebra11.1 Real Numbers
Objective: Students will be introduced to the real number system that is used throughout mathematics and will be acquainted with the symbols that represent them.
The Real Numbers
The real numbers can be ordered and represented in order on a number line
-3 -2 -1 0 1 2 3 4
-1.87
0
4.552
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
REAL NUMBERS (R)
Definition:
REAL NUMBERS (R)
- Set of all rational and
irrational numbers.
Definition:
REAL NUMBERS (R)
- Set of all rational and
irrational numbers.
SUBSETS of R
Definition:
RATIONAL NUMBERS (Q)
- numbers that can be expressed as a quotient a/b, where a and b are integers.
- terminating or repeating decimals
- Ex: {1/2, .25, 1.3, 5}
Definition:
RATIONAL NUMBERS (Q)
- numbers that can be expressed as a quotient a/b, where a and b are integers.
- terminating or repeating decimals
- Ex: {1/2, .25, 1.3, 5}
SUBSETS of R
Definition:IRRATIONAL NUMBERS (Q´)- infinite and non-repeating decimals- Ex: { ∏, √2, -1.436512…..}
Definition:IRRATIONAL NUMBERS (Q´)- infinite and non-repeating decimals- Ex: { ∏, √2, -1.436512…..}
SUBSETS of R
Definition:
INTEGERS (Z)
- numbers that consist of positive integers, negative integers, and zero,
- {…, -2, -1, 0, 1, 2 ,…}
Definition:
INTEGERS (Z)
- numbers that consist of positive integers, negative integers, and zero,
- {…, -2, -1, 0, 1, 2 ,…}
SUBSETS of R
Definition:
NATURAL NUMBERS (N)
- counting numbers
- positive integers
- {1, 2, 3, 4, ….}
Definition:
NATURAL NUMBERS (N)
- counting numbers
- positive integers
- {1, 2, 3, 4, ….}
SUBSETS of R
Definition:
WHOLE NUMBERS (W)
- nonnegative integers
- {0, 1, 2, 3, 4, …}
Definition:
WHOLE NUMBERS (W)
- nonnegative integers
- {0, 1, 2, 3, 4, …}
The Set of Real NumbersThe Set of Real Numbers
Q
Q'Q'QQ
ZZWW
NN
PROPERTIES of R
Definition:
CLOSURE PROPERTY
Given real numbers a and b,
Then, a + b is a real number (+),
or a x b is a real number (x).
Definition:
CLOSURE PROPERTY
Given real numbers a and b,
Then, a + b is a real number (+),
or a x b is a real number (x).
PROPERTIES of R
Example 1:
12 + 3 is a real number. Therefore, the set of reals is CLOSED with respect to addition.
Example 1:
12 + 3 is a real number. Therefore, the set of reals is CLOSED with respect to addition.
PROPERTIES of R
Example 2:
12 x 4.2 is a real number. Therefore, the set of reals is CLOSED with respect to multiplication.
Example 2:
12 x 4.2 is a real number. Therefore, the set of reals is CLOSED with respect to multiplication.
PROPERTIES of R
Definition:
COMMUTATIVE PROPERTY
Given real numbers a and b,
Addition: a + b = b + a
Multiplication: ab = ba
Definition:
COMMUTATIVE PROPERTY
Given real numbers a and b,
Addition: a + b = b + a
Multiplication: ab = ba
PROPERTIES of R
Example 3:
Addition:
2.3 + 1.2 = 1.2 + 2.3Multiplication:
(2)(3.5) = (3.5)(2)
Example 3:
Addition:
2.3 + 1.2 = 1.2 + 2.3Multiplication:
(2)(3.5) = (3.5)(2)
PROPERTIES of R
Definition:
ASSOCIATIVE PROPERTY
Given real numbers a, b and c,
Addition:
(a + b) + c = a + (b + c)
Multiplication: (ab)c = a(bc)
Definition:
ASSOCIATIVE PROPERTY
Given real numbers a, b and c,
Addition:
(a + b) + c = a + (b + c)
Multiplication: (ab)c = a(bc)
PROPERTIES of R
Example 4:
Addition:
(6 + 0.5) + ¼ = 6 + (0.5 + ¼) Multiplication:
(9 x 3) x 4 = 9 x (3 x 4)
Example 4:
Addition:
(6 + 0.5) + ¼ = 6 + (0.5 + ¼) Multiplication:
(9 x 3) x 4 = 9 x (3 x 4)
PROPERTIES of R
Definition:
DISTRIBUTIVE PROPERTY of MULTIPLICATION OVER ADDITION
Given real numbers a, b and c,
a (b + c) = ab + ac
Definition:
DISTRIBUTIVE PROPERTY of MULTIPLICATION OVER ADDITION
Given real numbers a, b and c,
a (b + c) = ab + ac
PROPERTIES of R
Example 5:4.3 (0.11 + 3.02) = (4.3)(0.11) + (4.3)(3.02)
Example 6:
2x (3x – b) = (2x)(3x) + (2x)(-b)
Example 5:4.3 (0.11 + 3.02) = (4.3)(0.11) + (4.3)(3.02)
Example 6:
2x (3x – b) = (2x)(3x) + (2x)(-b)
PROPERTIES of R
Definition:
IDENTITY PROPERTY
Given a real number a,
Addition: 0 + a = a
Multiplication: 1 x a = a
Definition:
IDENTITY PROPERTY
Given a real number a,
Addition: 0 + a = a
Multiplication: 1 x a = a
PROPERTIES of R
Example 7:
Addition:
0 + (-1.342) = -1.342 Multiplication:
(1)(0.1234) = 0.1234
Example 7:
Addition:
0 + (-1.342) = -1.342 Multiplication:
(1)(0.1234) = 0.1234
PROPERTIES of R
Definition:
INVERSE PROPERTY
Given a real number a,
Addition: a + (-a) = 0
Multiplication: a x (1/a) = 1
Definition:
INVERSE PROPERTY
Given a real number a,
Addition: a + (-a) = 0
Multiplication: a x (1/a) = 1
PROPERTIES of R
Example 8:
Addition:
1.342 + (-1.342) = 0 Multiplication:
(0.1234)(1/0.1234) = 1
Example 8:
Addition:
1.342 + (-1.342) = 0 Multiplication:
(0.1234)(1/0.1234) = 1
Inequality Graph Interval
3 7x
5x
1
3x
3,7
5,
1,
3
]
( ]
(5
3 7
1
3
) or ( means not included in the solution
] or [ means included in the solution
Inequalities, graphs, and notation
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
IntervalsInterval Graph
( )
[ ]
( ]
[ )
(
)
[
]
a b
Example
(a, b)
[a, b]
(a, b]
[a, b)
(a, )
(- , b)
[a, )
(- , b]
(3, 5)
[4, 7]
(-1, 3]
[-2, 0)
(1, )
(- , 2)
[0, )
(- , -3]
( )
[ ]
( ]
[ )
(
)
[
]
a b
a b
a b
a
a
b
b
3 5
-2 0
4 7
-1 3
-3
2
1
0
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Absolute Value
if 0
if 0
a aa
a a
To evaluate:
3 8 ( 5) 5 5Notice the opposite sign
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Real Number Venn Diagram
1-C
Scientific Notation
A short-hand way of writinglarge numbers without writing all of the zeros.
When using Scientific Notation, there are two kinds of exponents:
positive and negative
Positive Exponent:
2.35 x 108
Negative Exponent:3.97 x 10-7
An easy way to remember this is:
• If an exponent is positive, the number gets larger, so move the decimal to the right.
• If an exponent is negative, the number gets smaller, so move the decimal to the left.
The exponent also tells how many spaces to move the decimal:
4.08 x 103 = 4 0 8
In this problem, the exponent is +3, so the decimal moves 3 spaces to the right.
The exponent also tells how many spaces to move the decimal:
4.08 x 10-3 = 4 0 8
In this problem, the exponent is -3, so the decimal moves 3 spaces to the left.
Try changing these numbers from Scientific Notation to Standard
Notation:1) 9.678 x 104
2) 7.4521 x 10-3
3) 8.513904567 x 107
4) 4.09748 x 10-5
96780
.0074521
85139045.67
.0000409748
When changing from Standard Notation to Scientific Notation:
1) First, move the decimal after the first whole number:
3 2 5 8
123
3
2) Second, add your multiplication sign and your base (10).
3 . 2 5 8 x 10
3) Count how many spaces the decimal moved and this is the exponent. 3 . 2 5 8 x 10
When changing from Standard Notation to Scientific Notation:
4) See if the original number is greater than or less than one.– If the number is greater than one, the exponent
will be positive.
348943 = 3.489 x 105
– If the number is less than one, the exponent will be negative.
.0000000672 = 6.72 x 10-8
Try changing these numbers from Standard Notation to Scientific
Notation:1) 9872432
2) .0000345
3) .08376
4) 5673
9.872432 x 106
3.45 x 10-5
8.376 x 102
5.673 x 103
1-1 Answers (2-40e, 50,52)• 2. -,-,+,+• 4. >,<,=• 6. <,>,>• 8. b > 0, s < 0, w > -4, 1/5< c < 1/3, p < -2, -m > -2, r/s ≥ 1/5, 1/f ≤ 14, |x| < 4• 10. 10, 3, 17• 12. 4, 5/2, 10• 14. √3 -1.7, √3 – 1.7, 2/15• 16. 4,6, 6, 10• 18. 12, 3, 3, ,9• 20. | -√2-x|> 1 • 22. |4-x | < 2• 24. |x + 2| > 2 • 26. x – 5 • 28. 7 + x • 30. a – b • 32. x2 + 1• 34. =• 36 ≠• 38 ≠• 40 ≠• 50. 8.52 x 104 5.5 x 10-6 2.49 x 107
• 52. 23,000,000 .00000000701 12,300,000,000
1.2 Laws of Exponents
m n m na a a
Law Example
nm mna am
m nn
aa
a
n n nab a bn n
n
a a
b b
3 12 3 12 15x x x x
65 5(6) 303 3 3 14
14 12 212
yy y
y
4 4 4 43 3 81r r r 3 3
3 3
4 4 64
x x x
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Exponents
na 35 5 5 5 125 ...na a a a a
Definition
n factors
Examplen,m positive integers
0a
na
0 1 0a a
10n
na a
a
032 1
44
1 12
162
/m na
/m na
/ nm n ma a
/ 1m n
n ma
a
32 / 3 2125 125 25
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
8
27
4
9
9
432/3
http://www.youtube.com/watch?v=QIZTruxt2rQ&feature=related
1.2 Answers: p. 29 (12-30 x3)
12. -12x2
18.
24. -4x12y7
30. -288r8s11
5
12
y
Definitions
The entire expression, including the radical sign and radicand, is called the radical expression.
." ofroot square" theread is xx
x
thecalled issign radical theinside expression The
radicand.
thecalled is The radical sign.
Definitions
The positive or principal square root of a positive number a is written as . The negative square root is written as - .
aa
abba 2 if .00written0, is 0 ofroot square theAlso,
Note that the principal square root of a positive number, a, is the positive number whose square equals a. Whenever the term ‘square root’ is used in this book, the positive or principal square root is meant to be used.
Definitions
The index tells the “root” of the expression. Since square roots have an index of 2, the index is generally not written in a square root.
2 means xxExample:
25) 555 (since 525 2
)16
9
4
3
4
3
4
3 (since
4
3
16
92
Definitions
Square roots of negative numbers are not real numbers. Square roots of negative numbers are called imaginary numbers.
?25 There is no number multiplied by itself that will give you –25.
(Imaginary numbers will be discussed in a later section)
Cube and Fourth Roots
is read “the cube root of a.”3 a4 a is read “the fourth root of a.”
abba 33 if abba 44 if
8222 since 283
8)2)(2)(2( since 283
8133333 since 318 44
Even and Odd Indices
Even Indices
The nth root of a, , where n is an even index and a is a nonnegative real number, is the nonnegative real number b such that bn = a.
n a
813 since 381 44
10010 since 10100 2
Even and Odd Indices
Odd Indices
The nth root of a, , where n is an odd index and a is a any real number, is the real number b such that bn = a.
n a
644 since 464 33
32(-2) since 232 55
Cube and Fourth Roots
Note that the cube root of a positive number is a positive number and the cube root of a negative number is a negative number.
The radicand of a fourth root (or any even root) must be a nonnegative number for the expression to be a real number.
Evaluate by Using Absolute Value
For any real number a,
aa 2
7772
99)9( 2
119)119( 2 baba
)6()6()3612( 22 xxxx
Changing a Radical Expression
When a is nonnegative, n can be any index.When a is negative, n must be odd.
nn aa1
77 21
A radical expression can be written using exponents by using the following procedure:
3143 4 yxyx
9149 4 7373 zxzx
Changing a Radical Expression
When a is nonnegative, n can be any index.When a is negative, n must be odd.
nn aa1
15 15 21
Exponential expressions can be converted to radical expressions by reversing the procedure.
331 bb
Simplifying Radical Expressions
73 2372 9898 yxyx
This rule can be expanded so that radicals of the form can be written as exponential expressions. n ma
For any nonnegative number a, and integers m and n,
nmmnn m aaa
Power
Index
3 232 bb
Definitions
A perfect square is the square of a natural number. 1, 4, 9, 16, 25, and 36 are the first six perfect squares.Variables with exponents may also be perfect squares. Examples include x2, (x2)2 and (x3)2.
A perfect cube is the cube of a natural number. 1, 8, 27, 64, 125, and 216 are the first six perfect cubes.Variables with exponents may also be perfect cubes. Examples include x3, (x2)3 and (x3)3.
Perfect Powers
A quick way to determine if a radicand xn is a perfect power for an index is to determine if the exponent n is divisible by the index of the radical.
Example: 5 20x Since the exponent, 20, is divisible by the index, 5, x20 is a perfect fifth power.
This idea can be expanded to perfect powers of a variable for any radicand.The radicand xn is a perfect power when n is a multiple of the index of the radicand.
Product Rule for Radicals
Examples:
3333 424832
4444 3231648
3333 252125250
and numbers real enonnegativFor nnn abba
ba
,
Product Rule for Radicals
1. If the radicand contains a coefficient other than 1, write it as a product of the two numbers, one of which is the largest perfect power for the index.
2. Write each variable factor as a product of two factors, one of which is the largest perfect power of the variable for the index.
3. Use the product rule to write the radical expression as a product of radicals. Place all the perfect powers under the same radical.
4. Simplify the radical containing the perfect powers.
To Simplify Radicals Using the Product Rule
Product Rule for Radicals
Examples:
2623623672
4 354 34 204 3204 23 || bbbbbbb
3233 633 63 222816 xyyxyx
4 324 28164 3118 21632 yxyxyx 4 3274 2||2 yxyx *When the radical is simplified, the
radicand does not have a variable with an exponent greater than or equal to the index.
Quotient Rule for Radicals
Examples:
10
9
100
81
100
81
0 ,
and numbers real enonnegativFor
bb
a
b
a
ba
nn
n
,
3
75 5251
25Simplify radicand, if possible.
Quotient Rule for Radicals
More Examples:
4
2
3 12
3 6
312
6 46464
y
x
y
x
y
x
2
42
4 8
44 8
4 8
4 8
48
8
4132
56
2
3
16
3
16
3
16
3
16
3
b
a
b
a
b
a
b
a
ba
ba
241
216
1
32
2
64
2
64 22
3
5
3
5
xxx
x
x
x
x
CAUTION!
The product rule does not apply to addition or subtraction!
baba
baba
Rationalizing Denominators
Examples:
To Rationalize a Denominator
3
6
3
3
3
2
3
2
233
32
3
3
3
2
3
2 ||
y
yx
y
yxy
y
yx
y
y
y
x
y
x
Multiply both the numerator and the denominator of the fraction by a radical that will result in the radicand in the denominator becoming a perfect power.
r
prq
r
rpq
r
r
r
pq
r
pq
2
10
2
10
2
2
2
5
2
5 2444
Cannot be simplified further.
Conjugates
When the denominator of a rational expression is a binomial that contains a radical, the denominator is rationalized. This is done by using the conjugate of the denominator. The conjugate of a binomial is a binomial having the same two terms with the sign of the second term changed.
The conjugate of 65 is 65
The conjugate of 44 23 is 23 yxyx
12
125
12
12
12
5
)(
Simplifying Radicals
Simplify by rationalizing the denominator:
12
5
dc
dc
2
dc
dcdcdc
dcdc
dcdc
22
))((
))(2(
dc
dc
dc
dc 2
Simplifying Radicals
A Radical Expression is Simplified When the Following Are All True
1. No perfect powers are factors of the radicand and all exponents in the radicand are less than the index.
2. No radicand contains a fraction.
3. No denominator contains a radical.
Assignment:
• Day 2: Continued…pp. 29-31 (3-9, 33-81 x3, 92, 101/102)
• Day 3: Continued…pp. 29-31 (3-9, 33-81 x3, 92, 101/102)
Even Answers: Day 2: Continued… pp. 29-31 (3-9, 33-81 x3, 92, 101/102)
4. ½ 92. (a) -1.0813
6.5/1 (b) -44.3624
8.243/1
36.4r5/6
42.-y11/2
48.x5/3
54.a) 4+ x √x b) (4+x) √(4+x)
60.4
66.3 r s2 4√r
72.xy3 /5 • 4√5x2
78. 5x2 y5 √2
1.3 Algebraic Expressions
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Polynomials• Addition
3 2 33 2 7 15 5 13 12x x x x x 3 2 3
3 2
3 2 7 15 5 13 12
8 2 6 27
x x x x x
x x x
Combine like terms
• Subtraction
3 2 3 26 1 3 2x x x x x x 3 2 3 2
3
6 1 3 2
2 4 1
x x x x x x
x x
Combine like terms
Distribute
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Polynomials• Multiplication
2 5 3 2x x
Combine like terms
Distribute2 (3 2) 5(3 2)x x x
Distribute26 4 15 10x x x 26 11 10x x
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Polynomials
• Division
http://www.youtube.com/watch?v=uERRlY-WmmU
1.3 (4-44 x 4) Answers Day 1
4.
8.
12.
16.
20.
24.
28.
32.
36.
40.
44.
xxx 2106 23 22 15234 yxyx
24424117 234 xxxx
5103102 23456 xxxxxx
yxz 26629 yx
22 164025 yxyx 4224 2 yyxx
yx 3223 6414410827 yxyyxx
yzxzxyzyx 126494 222
1.3 Factoring Polynomials
3 26 36t t
• Greatest Common Factor
• Grouping
26 6t t
2 2 2mx mx x
1 2 1mx x x
The terms have 6t2 in common
2 1mx x
Factor mx Factor –2
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Factoring Polynomials
• Sum/Difference of Two Cubes:
• Difference of Two Squares:
2 9m
38 1x 22 1 4 2 1x x x
3 3m m
2 2x y x y x y
3 3 2 2x y x y x xy y
Ex.
Ex.
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Factoring Polynomials• Trinomials
2 5 6x x
3 26 27 12x x x
3 2x x
Ex.
Ex.
Trial and Error
23 2 9 4x x x
Trial and Error 3 2 1 4x x x
Greatest Common Factor
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
http://www.youtube.com/watch?v=OFSrINhfNsQ
POLYNOMIAL FUNCTIONS
The DEGREE of a polynomial in one variable is the greatest exponent of its variable.
A LEADING COEFFICIENT is the coefficient of the term with the highest degree.
What is the degree and leading coefficient of 3x5 – 3x + 2 ?
POLYNOMIAL FUNCTIONS
A polynomial equation used to represent a function is called a POLYNOMIAL FUNCTION.
Polynomial functions with a degree of 1 are called LINEAR POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 2 are called QUADRATIC POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 3 are called CUBIC POLYNOMIAL FUNCTIONS
1.3 Answers (48-100 x 4) Day 2
48.52.56.60.64.68.72.76.80.84.88.92.96.100.
)32(5 yxy )5711(11 232 rsrssr
Irreducible
)27)(53( xx2)74( z
)49)(49( trtr )5)(5( xxx
)34)(34(4 yxyx )253036)(56( 2363 yyxxyx
)2)(3( xyxay )42)(3)(2( 2 xxxx
)2)(2)(4( 224 xxx
)23)(23( xyxy 2)12( xx
1.4 Rational ExpressionsP, Q, R, and S are polynomials
Addition
Operation
Multiplication
Subtraction
Division
P Q P Q
R R R
P Q P Q
R R R
P Q PQ
R S RS
P Q P S PS
R S R Q RQ
Notice the common denominator
Reciprocal and Multiply
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Rational Expressions• Simplifying
2
2
25
7 10
x
x x
5 5
2 5
x x
x x
Cancel common factorsFactor
• Multiplying
2 2
3 2
2 1 6 6
1
x x x x
x x
3
1 1 6 1
1 1
x x x x
x xx
FactorCancel common factors
2
Multiply Across
5
2
x
x
2
6 1x
x
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Rational Expressions• Adding/Subtracting
3 2
4x x
Combine like terms
3 4 2
( 4) 4
x x
x x x x
Must have LCD: x(x + 4)
3 12 2 5 12
( 4) 4
x x x
x x x x
Distribute and combine fractions
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Other Algebraic Fractions• Complex Fractions
32
94
x
xx
Simplify to get to here
Distribute and reduce to get here
32
94
xx
x xx
2
3 2
9 4
x
x
Multiply by the LCD: x
3 2 1
3 2 3 2 3 2
x
x x x
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Other Algebraic Fractions
• Rationalizing a Denominator
7
3 y
Simplify
7 3
3 3
y
y y
21 7
9
y
y
Multiply by the conjugate
Notice: a b a b a b
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
1.4 Answers (3-30 x 3)
6.
12.
18.
24.
30.
23
12
x
x
)2(
12 xx
2
2
)25(
425
s
ss
x
x )13(2
)5(
10222
uu
uu
1.4 Answers Day 2 (33-51 x 3)
36.
42.
48.
22 sr
rs
ax
2
)32)(322(
10
xhx
Answers to Ch. Review1. Positive 15. 2. 84 16. 3. 6-x 17. 4. 3.865 x 102
5. 0.000093 6. 1.76 x 1013
7. 4x2y4
8. 9. 10. 11. 17x3 - 6x + 312. 12x3 + 73x2 + 79x – 5213. x4 + 13x2 – 1414. 64x3 + 336x2y + 588xy2 + 343y3
Simplifying Radicals Video
http://www.youtube.com/watch?v=pZSuMBXzEic
Complex Fractions Video
• http://www.wonderhowto.com/how-to-simplify-complex-fractions-algebra-365934/
Negative Exponents Video
http://www.youtube.com/watch?v=c4aiYf3fzVQ
Rational Expressions Video
http://www.youtube.com/watch?v=L1KD-C0lWsY