advanced math chapter p1 prerequisites advanced math chapter p
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Advanced Math Chapter P 1
Prerequisites
Advanced Math
Chapter P
Advanced Math Chapter P 2
Review of Real Numbers and Their Properties
Advanced Math
Section P.1
Advanced Math Chapter P 3
Natural numbers
• {1, 2, 3, 4, …}
Advanced Math Chapter P 4
Whole numbers
• {0, 1, 2, 3, 4, …}
Advanced Math Chapter P 5
Integers
• { … , -3, -2, -1, 0, 1, 2, 3, … }
Advanced Math Chapter P 6
Rational numbers
• Can be written as the ratio p/q where q ≠ 0• Includes natural, whole, integers, and
fractions.• The decimal representation of a rational
number either terminates (like 0.25) or is repeating.
Advanced Math Chapter P 7
Irrational numbers
• Are not rational• Have infinite non-repeating decimal
representations.
2
Advanced Math Chapter P 8
You try• Which of the numbers above are…• Natural numbers?
• Whole numbers?• Integers?• Rational numbers?• Irrational numbers?
1 6 1, , , 2, 7.5, 1.8, 22
3 3 2
1, 22
6, 22
3
6
3
1 6, , 7.5, 1.8, 22
3 3
6
3
Advanced Math Chapter P 9
Real numbers
• Used in everyday life to describe quantities• Includes rational and irrational numbers• Doesn’t include imaginary numbers
Advanced Math Chapter P 10
Real number line
• Numbers to the right of origin are positive, numbers to the left are negative
• Nonnegative numbers are positive or zero• Nonpositive numbers are negative or zero
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Origin
Advanced Math Chapter P 11
One-to-one correspondence
• Between real numbers and points on the real number line
• Every real number corresponds to one point on the number line
• Every point on the number line corresponds to one real number
Advanced Math Chapter P 12
Definition of order
• If a and b are real numbers, a is less than b if b – a is positive and on a number line, a is left of b
Advanced Math Chapter P 13
Bounded intervals
• Have endpoints• Have finite length• See chart on page 3
Advanced Math Chapter P 14
Closed intervals
• Include endpoints• Shown with square brackets• “or equal to”• Open intervals don’t include endpoints
(shown with parentheses)
Advanced Math Chapter P 15
Open intervals
• Don’t include endpoints• Shown with parentheses
Advanced Math Chapter P 16
Example
• Graph the following on a number line
3 5x
Advanced Math Chapter P 17
You try
• Graph the following on a number line
1 3x
Advanced Math Chapter P 18
Unbounded intervals
• Do not have a finite length• See chart on page 4
Advanced Math Chapter P 19
Example
• Express the following using inequality notation
• All x in the interval (–2,4]
2 4x
Advanced Math Chapter P 20
You try
• Express the following using inequality notation
• t is at least 10 and less than 22
10 22t
Advanced Math Chapter P 21
Absolute value
• Magnitude• Distance between the origin and the point
on the number line
if 0
if 0
a a a
a a a
Advanced Math Chapter P 22
Properties of Absolute values
• Chart on page 5
Advanced Math Chapter P 23
Distance between a and b
,d a b b a a b
Advanced Math Chapter P 24
Variables
• Letters used to represent numbers
Advanced Math Chapter P 25
Algebraic expressions
• Combinations of letters and numbers
13y 3 2z
24x
Advanced Math Chapter P 26
Terms
• Parts of an algebraic expression separated by addition (or subtraction)
Advanced Math Chapter P 27
Constant term
• Term that doesn’t contain a variable
Advanced Math Chapter P 28
Evaluating algebraic expressions
• Substitute numerical values for each of the variables in the expression
Advanced Math Chapter P 29
You try
• Evaluate the following for x = 12 5 4x x
Advanced Math Chapter P 30
Substitution Principle
• If a = b, then a can be replaced by b in any expression involving a.
Advanced Math Chapter P 31
Charts
• Pages 6, 7, and 8
Advanced Math Chapter P 32
You try
• Exercises 98 – 104 even
Advanced Math Chapter P 33
Factors
• If a, b, and c are integers such that ab = c, then a and b are factors, or divisors, of c.
Advanced Math Chapter P 34
Prime number
• Integer that has exactly two factors: 1 and itself
Advanced Math Chapter P 35
Composite
• Can be written as the product of two or more prime numbers
Advanced Math Chapter P 36
Fundamental Theorem of Artihmetic• Every positive integer greater than 1 can be
written as the product of prime numbers in precisely one way
• Prime factorization
Advanced Math Chapter P 37
Exponents and Radicals
Advanced Math
Section P.2
Advanced Math Chapter P 38
Exponential notation
• a to the nth power• n is the exponent• a is the base
na a a a a
Advanced Math Chapter P 39
Properties of exponents
• Chart page 12• Read first two paragraphs on page 13
Advanced Math Chapter P 40
Examples
• No calculator
5
2
5
5
2
4
3
3
Advanced Math Chapter P 41
You try
• No calculator
033 23
Advanced Math Chapter P 42
Example
• Rewrite with positive exponents and simplify
222x
Advanced Math Chapter P 43
You try
• Rewrite with positive exponents and simplify
2 44 8y y
Advanced Math Chapter P 44
Scientific notation
• n is an integer• Positive exponents mean large numbers• Negative exponents mean small numbers
10nc
1 10c
Advanced Math Chapter P 45
Examples
• Write in scientific notation• 9,460,000,000,000• 0.00003937
Advanced Math Chapter P 46
You try
• Write in scientific notation• 0.0000899• 34,000,000
Advanced Math Chapter P 47
You try
• Write in decimal notation• 1.6022 × 10-19
Advanced Math Chapter P 48
Definition of nth root
• Page 15
Advanced Math Chapter P 49
Principal nth root
• Page 15
Advanced Math Chapter P 50
Tables
• Page 16
Advanced Math Chapter P 51
Examples
• No calculators
503
4
Advanced Math Chapter P 52
You try
• No calculators
2
10 12 3
Advanced Math Chapter P 53
A radical is simplified when
• All possible factors have been removed from the radical
• All fractions have radical-free denominators• The index of the radical is reduced
Advanced Math Chapter P 54
Examples
8 3 54
• No calculators
Advanced Math Chapter P 55
You try
• No calculators
316
2775
4
Advanced Math Chapter P 56
Combining radicals
• Can add or subtract if they are like radicals• Have the same index and same radicand
• Should simplify first
Advanced Math Chapter P 57
Example
5 9x x
Advanced Math Chapter P 58
You try
• No calculators
4 27 75
Advanced Math Chapter P 59
Rationalizing denominators
• Gets rid of radical in denominator• Multiply both numerator and denominator
by the conjugate of the denominator
Advanced Math Chapter P 60
Conjugates
and a b m a b m
Advanced Math Chapter P 61
Examples
5
10
3
5 6
Advanced Math Chapter P 62
You try
3
7
6
2 3
Advanced Math Chapter P 63
Rationalizing numerators
• Sometimes useful• Not simplifying radical• Multiply numerator and denominator by
conjugate of numerator
Advanced Math Chapter P 64
Rational exponents
• Definition page 19
Advanced Math Chapter P 65
You try
• Change from radical to rational exponent form
3 64
9
Advanced Math Chapter P 66
You try
• Change from rational exponent form to radical form
12144
5416
Advanced Math Chapter P 67
You Try
• Simplify:
512 2
3 2
5 5
5
x
x
4 2x 24 3
Advanced Math Chapter P 68
Polynomials and Special Products
Advanced Math
Section P.3
Advanced Math Chapter P 69
Polynomial
• an is the leading coefficient
• n is the degree of the polynomial• A0 is the constant term
11 1 0
n nn na x a x a x a
Advanced Math Chapter P 70
Example
• Coefficients are 3, 7, 8, and -5• Leading coefficient is 3• Polynomial degree 4
4 23 7 8 5x x x
Advanced Math Chapter P 71
Polynomials in two variables
• Degree of each term is sum of exponents• Degree of polynomial is highest degree of
its terms• leading coefficient goes with highest-degree
term
25 6 4 9xy y xy
Advanced Math Chapter P 72
Standard form
• Written with descending powers of x, then descending powers of y
Advanced Math Chapter P 73
Adding and subtracting polynomials• Add or subtract like terms (have the same
variables to the same powers) by adding and subtracting their coefficients
Advanced Math Chapter P 74
You try
2 3 215 6 8 14.7 17x x x
Advanced Math Chapter P 75
FOIL
• ONLY FOR MULTIPLYING TWO BINOMIALS
• Product of first terms + Product of outside terms + Product of inside terms + Product of last terms
Advanced Math Chapter P 76
You try 3 2 2 8x x
Advanced Math Chapter P 77
Multiplying other polynomials
• Use the distributive property• Add the products of each term of the first
polynomial times the second polynomial
Advanced Math Chapter P 78
Example
2 23 7 2 3x x x
Advanced Math Chapter P 79
Special Products
• Page 26
Advanced Math Chapter P 80
Factoring Polynomials
Advanced Math
Section P.4
Advanced Math Chapter P 81
Factoring
• Writing a polynomial as a product• Unless noted otherwise, you want factors
with integer coefficients• Completely factored when each of its
factors is prime
Advanced Math Chapter P 82
Removing a common factor
• Distributive property in reverse• First step in factoring a polynomial
Advanced Math Chapter P 83
Example32 6x x
Advanced Math Chapter P 84
You try
23 4 3x x
Advanced Math Chapter P 85
Factoring special polynomial forms• Page 34• Come from special product forms in section
P.3
Advanced Math Chapter P 86
Examples23 27x
28 8 2t t
3 8x
Advanced Math Chapter P 87
You try2 64x
23 24 48t t
38 1x
Advanced Math Chapter P 88
Trinomials with binomial factors
• FOIL in reverse• May involve trial and error
Advanced Math Chapter P 89
Examples
2 2x x
23 5 2x x
Advanced Math Chapter P 90
You try
2 5 6x x
29 3 2x x
Advanced Math Chapter P 91
Factoring by Grouping
• Sometimes works for polynomials with more than three terms• Sometimes several different options will work
• Can use to factor trinomials
Advanced Math Chapter P 92
Examples3 22 6 3x x x
26 2x x
Advanced Math Chapter P 93
You try
3 26 2 3 1x x x
22 9 9x x
Advanced Math Chapter P 94
Rational Expressions
Advanced Math
Section P.5
Advanced Math Chapter P 95
Domain
• The set of real numbers for which an algebraic expression is defined
• Usually all real numbers, except any • that make the expression equal an
imaginary number • Or make it undefined
Advanced Math Chapter P 96
Examples
• Find the domain of the following:
5
2
x
x
3 3x
Advanced Math Chapter P 97
You try
• Find the domains of the following:
3
2 1
x
x
2 2x
Advanced Math Chapter P 98
Simplifying Rational expressions
• Factor each polynomial completely• Divide out common factors• List the domain by the simplified
expression• The domain of the simplified expression cannot
include numbers that weren’t in the domain of the original expression
Advanced Math Chapter P 99
Examples
• Write the following in simplest form215
10
x
x2 16
4
y
y
3 2
3
2 3
1
y y y
y
Advanced Math Chapter P 100
You try
• Write the following in simplest form36
2
x
x2 5 6
2
y y
y
3 2
2
2 2
1
x x x
x
Advanced Math Chapter P 101
Operations with rational expressions• Factor• Then multiply, divide, add, or subtract using
the rules for fractions
Advanced Math Chapter P 102
Examples
5 1
1 25 2
x
x x
2 2
2
6 4
6 9 3
t t t
t t t
2 3
1 2 1
1x x x x
Advanced Math Chapter P 103
You try2 2
3 2 2 2
2
3 2
x xy y x
x x y x xy y
3 5
2 2x x
2
2 2 1
1 1 1x x x
Advanced Math Chapter P 104
Complex fractions
• Have separate fractions in the numerator, denominator, or both.
• Two ways to solve• Making one fraction in numerator and one in
denominator and dividing• Multiply numerator and denominator by LCD
of all fractions involved
Advanced Math Chapter P 105
Example3
4 23
2
x
x
Advanced Math Chapter P 106
Example
2
2
1
1
xx
x
x
Advanced Math Chapter P 107
You try
12
2
x
x
Advanced Math Chapter P 108
Simplifying expressions with negative exponents• Factor out the common factor with the
smaller exponent• When factoring, subtract exponents
Advanced Math Chapter P 109
Example
5 42 2 21 1x x x
Advanced Math Chapter P 110
You try
3 422 5 4 5x x x x
Advanced Math Chapter P 111
Difference Quotient
• Have a difference on the top and a constant or degree 1 term on the bottom
• In calculus, sometimes you have to rewrite them by rationalizing the numerator so that the expression is defined.
Advanced Math Chapter P 112
Example
1 1x h x
h
Advanced Math Chapter P 113
Errors and the Algebra of Calculus
Advanced Math
Section P.6
Advanced Math Chapter P 114
Common algebraic errors
• Read lists on your own during homework time
• Ask if you don’t understand
Advanced Math Chapter P 115
Algebra of Calculus
• Sometimes writing things in an “unsimplified” way makes doing calculus operations easier
• Read on your own• Let me know if you have questions
Example
• Simplify the expression
Advanced Math Chapter P 116
4 35 2 2 4
25
3 1 2 1 5x x x x x
x
Example
• Write the fraction as the sum of three terms
Advanced Math Chapter P 117
2 4 8
2
x x
x
You try
• Write the fraction as the sum of three terms
Advanced Math Chapter P 118
22 1x x
x
Advanced Math Chapter P 119
The Cartesian Plane
Advanced Math
Section P.7
Advanced Math Chapter P 120
Cartesian Plane
• Rectangular coordinate system• Named after René Descartes• Ordered pair: (x, y)• Horizontal x-axis• Vertical y-axis• Origin: where axes intersect
Advanced Math Chapter P 121
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1
2
3
4
5
x
y
I
IVIII
II
Advanced Math Chapter P 122
Scatter plots
• Each point is plotted• Dots are not connected
Advanced Math Chapter P 123
Distance formula
2 222 1 2 1d x x y y
2 2
2 1 2 1d x x y y
2 2
2 1 2 1d x x y y
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(x1, y1)
(x1, y2) (x2, y2) x
y
• Pythagorean theorem
d
2 2 2a b c
Advanced Math Chapter P 124
You try
• Find the distance between (-3, 2) and (3, -2)
Advanced Math Chapter P 125
Verifying a right triangle
• Showing that three given points are vertices of a right triangle.
• Plot the points• Use the distance formula to find the
distances between the points.• See if the distances work in the Pythagorean
theorem
Advanced Math Chapter P 126
Example
• Use the distance formula to show that the points (9,4), (9,1), and (-1,1) form a right triangle.
Advanced Math Chapter P 127
Midpoint formula
• To find the midpoint of the line segment joining two points, average the x-coordinates and average the y-coordinates.
• Midpoint has coordinates
1 2 1 2,2 2
x x y y
Advanced Math Chapter P 128
Example
• Find the midpoint of the segment joining the points (1, 1) and (9, 7).
Advanced Math Chapter P 129
You try
• Find the midpoint of the line segment joining the points (–1, 2) and (5, 4).
Advanced Math Chapter P 130
Example
• Use the midpoint formula to find points that divide the line segment joining (1, –2) and (4, –1) into four equal parts.