MAC 1105-College Algebra LSCC, S. Nunamaker Chapter R-Basic Concepts of Algebra R.1 The Real Number System I. Real Number System Please indicate if each of these numbers is a W (Whole number), R (Real number), Z (Integer), I (Irrational number), N (Natural number), Q (Rational number) Examples: -6, 7, 54 ., - 3 7 , 8 3 , 6 242242224 . , 6 24224222422224 . ...., P, - 5 11 , 2, 19 3 , -0 2020020002 . .... Natural Numbers: {1, 2, 3, 4,....} Whole Numbers: {0, 1, 2, 3, 4, ....} Integers: {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ....} Rational Numbers: { p q p and q are integers and q „ 0} Irrational Numbers: { xx is real but not rational} Real Numbers: {x x corresponds to a point on a number line} II. Properties of Real Numbers For any real numbers a, b, and c: a + b = b + a a b = b a a + ( b + c ) = ( a + b ) + c associative property a ( b c ) = ( a b ) c associative property a + 0 = 0 + a = a identity property a -a = a + ( - a ) = 0 inverse property of addition 1
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MAC 1105-College Algebra LSCC, S. Nunamaker
Chapter R-Basic Concepts of Algebra
R.1 The Real Number System
I. Real Number System
Please indicate if each of these numbers is a W (Whole number), R (Real number), Z(Integer), I (Irrational number), N (Natural number), Q (Rational number)
III. Absolute Values: Distance on the number line from 0 to the number. Absolute value isalways positive. Absolute value of a is a
Example: 7 7= - =7 7
IV. Distance
Absolute value of the difference between two numbers
Examples: a. distance between -2 and -8
b. distance between - 4 and -14
c. distance between 12.1 and 5.1
d. distance between 15
8 and
23
12
e. distance between 16 and -8
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R.2 Integer Exponents, Scientific Notations, and Order of Operations LSCC,S. Nunamaker
I. Integers as Exponents
A. For any positive integer n, a a a a a an = × × × ××× n times
such that a is the base and n is the exponent
Example: a a a a3 = × × 3 3 3 33 = × × = 27
B. For any nonzero real number a and any integer n,
a0 1= and aa
n
n
- =1
Examples: 1. 60 =
2. ( )- =2 0
3. ( )- =2 3
4. -23 =
5. 4 2- =
II. Properties of Exponents
a a am n m n× = + ( )ab a bm m m=
( )a
b
a
bm
m
m= such that b ¹ 0
a
aa
m
n
m n= -( ) such that a ¹ 0
( )a am n mn=
Examples:
1. ( )2 2 5x - = 2. 45
15
8
2
x
x=
3. y y- × =5 2 4. x x2 5× =
5. y y- -¸ =6 3 6. y
y
2
4-=
3
7. ( )- =2 3 4x 8. y
y
-
=3
2 LSCC, S. Nunamaker
9. m m- × =5 5 10. ( )2 3 5x - =
11. ( )24
3
10 8 7
6 3 5
5a b c
a b c
-
-= 12. ( )
27
9
4 2
2 8
3x y
x z
-
-
- =
III. Scientific Notations
Scientific notation for a number is an expression of the type
N x 10m , such that 1 10£ <N , N is in decimal notation, m is an integer
In scientific notation, numbers appear as a number greater than or equal to 1 and less than10 multiplied by some power of 10.
Ex. 1. FM radio signal may be 14,200,000,000 hertz (cycles per second); in scientificnotation, this is 1.42 ́ 1010 hertz.
2. Diameter of an atom is 0.0000000001 meter,
In scientific notation, this is 1 10 10´ - meter
Try to express the result in scientific notation:
1. ( . )( . )91 10 82 1017 3´ ´ =- 2. 6 4 10
80 10
7
6
.
.
´
´
-
=
3. 145 000 000, , = 4. 000876. =
Try to express the following without exponents:
1. 23 104. ´ = 2. 897 10 5. ´ - =
3. 146 106. ´ = 4. 457 10 3. ´ - =
*In 2000, the number of people living in the world was about 609 109. ´ . The number of peoplelived in US at that time was about 2 74 108. ´ . How many people lived outside of US in 2000?
*We have proof that there are at least 1 sextillion, 1021 , stars in the Milky Way. Write thisnumber without the use of exponent.
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LSCC, S. Nunamaker
Examples:
1. Convert each of the followings to decimal notation:
a. 7654 10 5. ´ - = b. 6 45 105. ´ =
2. Convert each of the followings to scientific notation:
a. 876 420 000, , = b. 0000542. =
c. ( . )( . )91 10 82 1017 3´ ´ =- d. 6 4 10
80 10
7
6
.
.
´
´
-
=
e. 432 3 4 1012. ( . )´ ´ = f. 25 10
5 10
6
2
. ´
´=
g. 46 26 10
25 4 10
8
2
. .
.
´ ´
´ ´=
-
IV. Order of Operations
Please Excuse My Dear Aunt Sally
( ) a exp ́ ̧ + -
1. Exponential expression and calculations within grouping
symbols first and always from left to right.
2. Multiplication and division from left to right
3. Addition and subtraction from left to right
Examples:
1. ( )6 3 22 + = 2. 8 5 3 103( )- - =
3. - - - +4 3 2 42 2 2{ [ ( )]}a a b a 4. [ ( ) ]( )
Principal P is invested at an interest rate i, compounded n
times per year, in t years it will grow to an amount A given
by: A Pi
nnt= +( )1 or A P
r
nnt= +( )1
A: total amount, P: principal, i r= = annual interest rate
n: number of times compounded per year, t: number of years
Examples:
1. Suppose $9550 is invested at 5.4%, compounded semiannually. How much is in theaccount at the end of 7 years?
2. Suppose $6700 is invested at 4.5%, com- pounded quarterly. How much is in theaccount at the end of 6 years?
3. Suppose you will be needing $20,000 in ten years, how much do you need to investnow (principal), if the investment will be earning interest at 10% and compounding semi-annually?
4. Suppose you will be needing $20,000 in ten years, how much do you need to investnow {principal}, if the investment will be earning interest at 10% and compounding quarterly.
5. Suppose you will be needing $20,000 in ten years, how much do you need to investnow (principal), if the investment will be earning interest at 5% and compouding semi-annually.
6. The interest earned on an $800 investment at 71
4% annual interest compounded
monthly for 6 months is ?
7. Shane begins a new job with an annual salary of $40,000 and a guarantee of a 2%salary increase every year for the first 5 years. After that, he is given 3% increase each year. What will be his salary in 8 years?
8. A couple want to have $100,000 in 21 years for their newborn son. How much money should be deposited now in an account earning 6
1
2% annual interest compounded quarterly?
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R.3 Addition, Subtraction, and Multiplication of Polynomials LSCC, S. Nunamaker
I. Polynomials in One Variable
a x a x a x a x ann
nn+ + + + +-
-1
12
21 0....
and that n is a nonnegative integer,
and a an,..., 0 are real numbers called coefficients
and an ¹ 0
A. Definition
1. Terms
2. Degree of the polynomial
3. Leading coefficient
4. Constant term
5. Descending order
Examples:
a. 2 8 204 3x x x- + - b. y y2 31
26- +
6. Monomial
7. Binomial
8. Trinomial
II. Polynomial in Several Variables
A. Definition
1. Degree of a term - sum of the exponents of the variables in that term.
2. Degree of a polynomial - the degree of the term of highest degree.
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Examples: LSCC, S. Nunamaker
a. 9 12 93 2 4ab a b- +
b. 7 5 3 64 3 3 2 2x y x y x y- + +
III. Expressions That Are Not Polynomials
a. 2 552x xx
- + b. 20 - x c. y
y
+
+
1
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IV. Addition and Subtraction of Polynomials
Like terms - terms/expressions that have same variables
raised to the same powers.
Combine/collect like terms
Examples:
a. ( ) ( )- + - + - +5 3 12 7 33 2 3 2x x x x x
b. ( ) ( )8 9 6 32 3 2 3x y xy x y xy- - -
c. ( ) ( )3 2 2 5 8 42 3 2 3x x x x x x- - + - - - +
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V. Multiplication of Polynomials LSCC, S. Nunamaker
A. (binomial)(binomial) : binomial multiplied by binomial,