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Math 72 Course Pack
PAGE
38
Math 72 Course Pack
Fall 2010
Section 0249 ONLY
Instructor: Yolande Petersen
DO NOT BUY THESE NOTES IF YOU HAVE A DIFFERENT INSTRUCTOR
Inside:
· Lecture Notes Outline with writing space for your notes
· Syllabus
· Calendar of Material Coverage and Exam Dates
· Homework Assignments for each section
· Cumulative Review Problems (sample test problems, Petersen style only)
· Final Exam Review
Suggestions for use
The lecture notes pages are in reverse order, upside down, and punched on the "wrong" side for a reason! My notes read like a book, with the printed side on the left and handwritten extra notes on the right. If you want your notes look like mine, insert the lecture pages into a binder with the blank back of page 1 on top. When you turn the first page, page 1 will be on your left, with the blank back of page 2 on your right, where you can write additional notes. Effort was made to minimize the number of pages printed to reduce your cost, while leaving enough space for your notes to be arranged in an orderly way. If you don't like this arrangement, feel free to order the pages however you like. The supplements (syllabus, homework assignments, and review problems) can be inserted into a binder in the "usual" way.
Mrs. Petersen's website: http://www.virtual.mjc.edu/peterseny
Before you take this class, you may find it helpful to read the document at the link, "Teaching Style and Educational Philosophy" to decide whether this instructor is a good match for you.
Chapter 5 Review Concepts
Distributive Property (multiplying polynomials by monomials)
Ex a
FOIL (multiplying 2 binomials) – First, Outer, Inner, Last
Special Product Formulas
1. (f + s)2 = f2 + 2fs + s2
2. (f – s)2 = f2 – 2fs + s2
3. (f + s)(f – s) = f2 – s2
Common Errors:
6.1 The Greatest Common Factor (GCF); Factoring by Grouping
Factors - things multiplied to make a product
It is sometimes desirable to write things as factors (pieces)
Prime factored form - a product of factors where none of the factors can be "broken down" further (with exponents consolidated)
Greatest Common Factor (GCF) – the largest factor (small broken-down piece) included in all the numbers
Least Common Multiple (LCM) - the smallest multiple (large multiplier) that includes all the numbers
Finding the GCF
1. Write all the numbers in prime factored form
2. For each factor, choose the smallest exponent common to all the numbers
3. Write the product
Finding the LCM
1. Write all the numbers in prime factored form
2. For each factor choose the largest exponent possible
3. Write the product
Factoring out the GCF – the reverse of the distributive law
1. Find the GCF (by eyeballing or previous method) and write it in front
2. Divide the GCF out of each term
3. Write the "leftovers" inside parentheses
Factoring by Grouping – take out identical "clumps" of stuff
6.2 Factoring Trinomials (of form x2 + bx + c) – reverse of FOIL
x2 + bx + c factors to (x + ?)(x + ?)
How do we know whether it's + or - , and how do we get ?
Some examples of FOIL
Factored
Unfactored
(x + 1)(x + 2) = x2 + x + 2x + 2 =
x2 + 3x + 2
(x – 3)(x – 5) = x2 – 5x – 3x + 15 = x2 – 8x + 15
(x – 4)(x + 3) = x2 + 3x – 4x – 12 = x2 – x – 12
(x – 2)(x + 5) = x2 + 5x – 2x – 10 = x2 + 3x – 10
Observations on signs
1. If c is positive
Þ
a. If b is positive
Þ
b. If b is negative
Þ
2. If c is negative
Þ
a. If b is positive
Þ
b. If b is negative
Þ
Observations on numbers
1. c is the product of the 2 binomial numbers
Þ
2. If the 2 binomial numbers have the same sign, b
3. If the 2 binomial numbers have different signs, b
6.3 More Factoring Trinomials (ax2 + bx + c)
ax2 + bx + c factors to (?x + ?)(?x + ?)
Factoring by grouping – long process, no guesswork
1. Multiply
c
a
×
2. For c positive, find 2 factors whose sum = b
For c negative, find 2 factors whose difference = b
3. Rewrite the whole polynomial with the middle term written as the sum or diff.
4. Factor by grouping
Ex a
Trial and Error
· For simple numbers with few factors, it's much faster than grouping
· For numbers with many factors, requires much guesswork and luck
· Rules for signs are same as section 6.2
· Rules for numbers require "mix and match" to get the middle number, b
6.4 Special Factoring
Special formulas (same as before, written in reverse):
1. f2 – s2 = (f + s)(f – s)
2. f2 + 2fs + s2 = (f + s)2
3. f2 – 2fs + s2 = (f – s)2
Ask: Does my binomial or trinomial fit one of these 3 patterns
Binomials – Use 1st formula for squares, last 2 formulas for cubes
Trinomials - Use formulas 2 or 3
More Formulas – cubes
4. f3 + s3 = (f + s)(f2 – fs + s2)
5. f3 – s3 = (f – s)(f2 + fs + s2)
Common Errors:
x3 + y3
¹
(x + y)3
x3 – y3
¹
(x – y)3
Note: x2 – y2 can be factored
x2 + y2 can't be factored
Factoring Summary
1. Can a GCF be factored out? (6.1)
2. 2 terms: Is it (f2 – s2) or (f3 + s3) or (f3 – s3)? (6.4)
3. 3 terms: Is it (f2 + 2fs + s2) or (f2 – 2fs + s2)? (6.4)
Does trinomial
®
2 binomial factoring work easily? (6.2)
Is there a number in front of x2 for trinomial
®
2 binomials? (6.3)
4. 4 terms: Does factoring by grouping work?
5. Can any factors be further factored?
6.5 Solving Quadratic Equations – Factoring
Quadratic equation – 2nd degree
Standard form: ax2 + bx + c = 0
Zero Factor Property – basis for solving quadratic equations
If ab = 0, then a = 0 or b = 0 (one of the numbers MUST be zero)
Ex a
Procedure for solving
1. Write equation in standard form (get 0 on one side)
2. Factor
3. Set each piece (factor) equal to 0
4. Solve each piece; check
The number of solutions is equal to or less than the degree of the equation.
How what is the maximum number of solutions of a quadratic equation?
How many solutions could a cubic equation potentially have?
6.6 Applications of Quadratic Equations
Rectangles: A = LW; P = 2L + 2W
Triangles:
Pythagorean Theorem: In a right triangle with hypotenuse c and legs a & b,
a2 + b2 = c2
Ex The hypotenuse of a right triangle is 1 cm longer than the longer leg of the triangle. The shorter leg is 7 cm shorter than the longer leg. Find the length of the longer leg.
Numbers
Velocity/Distance
Ex The height of a ball with an initial velocity of 128 ft/sec which travels t seconds is described by the equation:
h = 128t – 16t2
a) What is the height of the ball after 2 seconds?
b) For what values of t is h = 0?
c) Interpret the values from b) in real life terms.
d) For what values of t is h = 240 ft?
e) After how many seconds do you think the ball will reach its peak?
7.1 The Fundamental Property of Rational Expressions (Canceling Fractions)
Rational Expression: Can be written as P/Q, where P and Q are polynomials , Q
¹
0
(Note: Q = 1, so non-fractions can be made into fractions)
P/Q is undefined when Q = 0.
A place where the denominator is undefined is called a
Ex a For what values is the expression
5)
1)(x
(x
3
x
-
+
-
undefined?
Lowest Terms – The rational expression P/Q is in lowest terms if there are no common factors in the denominator.
Fundamental Property of Rational Expressions (Canceling Rule)
Q
P
QK
PK
=
where K
¹
0
7.2 Multiplying and Dividing Rational Expressions
Multiplying:
QS
PR
S
R
Q
P
=
×
Smart Way: Cancel (reduce) as much as possible before multiplying
Procedure:
1. Factor
2. Cancel and rewrite the remaining factors
3. Note which bad points have disappeared (for extra credit)
Dividing:
QR
PS
R
S
Q
P
S
R
Q
P
=
×
=
¸
7.3 The Least Common Denominator (LCD)
Recall: The LCD is the least common multiple of the denominators. It is as large as or larger than each of the individual denominators.
Finding the LCD of Polynomial Factors
1. Factor each denominator completely, writing repeat factors in exponent form
2. Write each unique factor to the highest power possible
Caution: Don't confuse the LCD with the GCF.
Ex a
Building a fraction to match a denominator
1. Compare the new and old denominators, and find "what's missing" from the old denominator. When in doubt, divide the LCD by the old denominator.
2. Multiply top and bottom by the missing factor.
7.4 Adding and Subtracting Rational Expressions
Rules:
Q
R
P
Q
R
Q
P
+
=
+
and
Q
R
P
Q
R
Q
P
-
=
-
for Q
¹
0
Caution: 2 fractions must have same denominator. If not, a common denominator (LCD) must be used.
Procedure -- Adding/Subtracting with same denominator
1. Add or subtract numerators, keep denominator
2. Factor and/or cancel if possible
Adding/Subtracting with different denominators
1. Find the LCD
2. Build the fractions to match LCD
3. Add or subtract as above
7.5 Complex Fractions
· Typically have 4 layers (2 top, 2 bottom)
· We can treat 4 layers as 2 separate fractions, divided
2 Solving methods
Method 1: Invert bottom fraction and multiply
Method 2: Multiply both fractions by LCD to clear 2 denominators
Method 1 is generally more reliable; Method 2 can be fast but is prone to errors
Clearly Separated Layers – Use Method 1
Ex a
Partial Layers – Use either method
7.6 Equations with Rational Expressions
Recall:
Expressions
Equations
· no equal sign
· Goal: Keep denominator by putting numerator over LCD
· Final answer may have mixed variables and numbers
· Process is to simplify
· has equal sign
· Goal: get rid of denominator by multiply by LCD
· Final answer: x = number (variable is isolated)
· Process is to solve
Solving Procedure
1. Find LCD
2. Multiply both sides by LCD
3. Cancel factors to get rid of denominators
4. Solve
5. ***Check if solution is a bad point – This is now required, not optional
Solving for a Specified Variable
1. Clear denominators if necessary
2. Get all terms with desired variable on one side, all other terms on other side
3. Factor out desired variable
4. Divide by "junk"
7.7 Applications
Distance/Rate/Time
d = rt; isolating for different variables, r = d/t or t = d/r
Work/Rate/Time
w = rt or r = w/t or t = w/r
"3 hours to paint a room" can be interpreted in 2 ways:
1. t = 3 hours
2. r = 1 room/3 hours or 1/3 room/hour (this is most commonly used)
8.1 Evaluating Roots
square rooting - the reverse process of squaring
Suppose we are given x2 = 25. What is the "unsquare" of 25?
All square roots of "a"
If "a" is a positive, real number,
a
is the positive square root of "a"
-
a
is the negative square root of "a"
Note the difference between word form and symbol form:
1. "the square roots of x"
2.
x
Types of Roots
1. Rational Roots – If "a" is a perfect square, then
a
is rational, e.g.
2. Irrational Roots – If "a" is positive but not a perfect square,
a
is irrational
To get a ballpark idea of a number:
3. Non-real roots – If "a" is negative,
a
is non-real
Pythagorean theorem
c2 = a2 + b2 or
2
2
b
a
c
+
=
Higher roots – It's helpful to know some perfect squares, cubes, 4th powers, etc.
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
x2
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
x3
1
8
27
64
125
216
x4
1
16
81
256
625
x5
1
32
243
Example:
3
8
"The cube root of 8"
8.2 Multiplying, Dividing, Simplifying Radicals
Multiplying: Product Rule
xy
y
x
=
and
y
x
xy
=
Simplifying: A radical is completely simplified when no perfect square factor is inside the radical symbol.
2 Methods for simplifying:
· Method 1 – Find prime factorization & take out even powers
· Method 2 – "Eyeball" perfect squares that are easy to see & take them out until no perfect squares are left
Quotient Rule
y
x
y
x
=
and
y
x
y
x
=
Radicals with variables
For even powers inside square roots – divide power by 2
For odd powers inside square roots – separate off 1 power, divide the rest by 2
8.3 Adding & Subtracting Radicals
Don't confuse with multiplication:
y
x
y
x
+
¹
+
Radicals are similar to variable like terms – only like types can be added/subtracted
Higher roots
Only like radicals (same index and number under radical) can be combined
Goal: For cube roots, take out perfect cubes, e.g 8, 27, 64, 125 or x3 x6, x9…
For 4th roots, take out perfect 4th powers, e.g. 16, 81, 256 or x4, x8, x12…
8.4 Rationalizing the Denominator
Completely simplified radicals have:
1. No perfect squares inside radical sign
2. No fractions inside radical sign
3. No radicals in the denominator
Procedure for Rationalizing the Denominator
1. Simplify the denominator radical as much as possible
2. Examine the what remains inside the radical and determine "what's missing" to make a whole (non-radical)
3. Multiply top and bottom by the missing part
8.5 Simplifying Radical Expressions
Completely simplified radicals should have:
1. No parentheses (distribute or multiply out all terms)
2. No perfect square factors inside radical
3. No fractions inside radical or radicals in the denominator
4. Products of radicals should be written under one "roof"
5. Sums of like radicals should be combined
Multiplying – distributive law
Multiplying – FOIL
Conjugates – a pair of binomials that have 1) 1st terms exactly the same and 2) 2nd terms that are the same, except for opposite signs
Writing a radical in lowest terms
You CAN'T cancel terms (things added/subtracted) in the top & bottom
You CAN cancel factors (things multiplied) in top & bottom
Method 1: Factor top and bottom, then cancel
Method 2: Separate numerator terms, putting each over its own denominator, then cancel
8.6 Solving Equations with Radicals
Goal: Isolate the variable
Recall: Squaring is the reverse of square root, so squaring a radical "undoes" it
Procedure:
1. Isolate the radical. If there are 2 radicals, get one on each side
2. Square both sides. Combine like terms
3. If there is still a radical, isolate it. Repeat steps 1 & 2.
4. Solve for potential solutions
5. Check all potential solutions – this is mandatory!
8.7 Fractional Exponents
By definition:
a ½ =
a
a 1/3 =
3
a
a 1/n =
n
a
To verify:
Other fractional Exponents
a m/n =
(
)
m
n
n
m
a
a
=
a m/n = (am)1/n = (a 1/n)m
Observation on solutions:
Linear (first degree) equations typically have 1 solution
Quadratic (2nd degree) equations typically have 2 solutions
Radical equations (1/2 power) typically have solutions that are good "half the time"
Fractional Expressions
Recall:Product rule:
n
m
n
m
a
a
a
+
=
×
Quotient rule:
n
m-
n
m
a
a
a
=
Power rule:
mn
n
m
a
)
(a
=
9.1 Solving Quadratic Equations – Square Root Property
Recall: Quadratic equations look like ax2 + bx + c = 0 (2 solutions maximum)
Ex a Solve x2 – 5x = - 4
4 Common Methods of Solving Quadratic Equations
Method
Advantages
Disadvantages
1. Factoring
fast, simple
Doesn't solve every equation
2. Square Root
fast, simple
Doesn't solve every equation
3. Complete the Square
Solves every equation
Requires thinking
4. Quadratic Formula
Solves every equation
Tedious, requires many steps
Square Root Method – works when b – 0 (only have x2 and constant terms, no x term)
Square Root Property
If a2 = k, then a = +
k
or a = –
k
Remember: Quadratics may have
1. 2 solutions
®
x2 = positive number (or (stuff)2 = positive #)
2. no solution
®
x2 = negative number
3. one solution
®
x2 = 0
9.2 Completing the Square
Suppose we have x2 + 10x + 25 = 36
Procedure for Completing the Square
1. Get equation in the form x2 + bx = c (x2 and x terms on left, number on right)
2. Find the needed number to complete the square
nn =
2
÷
ø
ö
ç
è
æ
2
b
3. Add it to both sides.
4. Factor the perfect square. Write it as a square that looks like (x + b/2)2
5. Square root both sides
6. Solve for x
9.3 Quadratic Formula
Recall:
Quadratic equations have the form ax2 + bx + c = 0
Quadratic equations have 2, 1, or 0 solutions
Quadratic Formula
The equation ax2 + bx + c = 0 has solutions:
x =
2a
4ac
b
b
2
-
+
-
or x =
2a
4ac
b
b
2
-
-
-
Putting the 2 together:
x1,2 =
2a
4ac
b
b
2
-
±
-
Your book derives this formula on p. 608. We won't!
The inside of the square root is sometimes called the discriminant.
· If b2 – 4ac > 0
Þ
2 real roots
· If b2 – 4ac = 0
Þ
1 real root
· If b2 – 4ac < 0
Þ
no real root
9.4 Complex Numbers
The number i is defined as i =
1
-
(a non-real number)
i2 = -1
Complex numbers have the form
a + bi e.g. -3 + 2i or 7 - 4i
There are pure real, pure imaginary numbers, and numbers with a mix of each.
Imaginary numbers refers to any number with an imaginary part (either pure imaginary or a mix of real & imaginary parts).
Complex numbers include all 3 types (pure real, pure imaginary, mixed)
Adding and Subtracting – add/subtract real and imaginary parts separately, similar to like terms
Dividing
· Imaginary numbers are not allowed in the denominator
· Rationalize by multiplying top and bottom by the conjugate, similar to the way radicals in the denominator are eliminated.
Multiplying imaginary conjugates produces a real number:
(a + bi)(a - bi) = a2 – abi + abi – b2i2 = a2 – b2(-1) = a2 + b2
9.5 Graphing Quadratic Equations/Quadratic Functions
Recall some features of linear functions (1st degree):
1. They can be written as y = mx + b
2. They are straight lines
3. Important landmarks are m (slope) and b (y-intercept)
Some features of quadratic equations (2nd degree):
1. They can be written as y = ax2 + bx + c
2. They are
3. Important landmarks include:
Direction of opening
Vertex and axis of symmetry
Width (of parabola)
Finding the Vertex (better way than trial & error)
x-coordinate: x =
2a
b
-
EMBED Equation.3
y-coordinate: plug the value of x into the original equation to get y
Procedure for Graphing a 2nd degree equation (parabola)
Decide if the parabola opens up or down
Calculate the vertex
Plot at least 1 extra point to find the width of the parabola
Intercepts of a Graph of a Quadratic Function
A graph has its x-intercepts when y = f(x) = 0 (set equation = 0)
The x-values are the places where the graph crosses the line.