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Math 70 Course Pack
Beginning Algebra
Fall 2015
Instructor: Yolande Petersen
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Mrs. Petersen's website: http://peterseny.faculty.mjc.edu
Before you take this class, you may find it helpful to read the document "Teaching Style and
Educational Philosophy" to decide whether this instructor is a good match for you. You can
find it at the above web address, with the link on the home page under the "What To Expect"
heading.
1
1.1 Intro to Algebra: Expressions, Equations, Translating
Variables - needed for quantities that change
Evaluating Expressions – plug in a number for each variable
Ex a
Ex The area of a triangle is A = ½ bh. Find the area if the base is 6 inches and height is 9
inches.
Translating English to Algebra
4 important words:
sum
difference
product
quotient
Addition
the sum of a number and 2
5 more than a number
3 added to t
Subtraction
the difference of 3 and a number
3 less than y
7 subtracted from a number
x decreased by 8
Multiplication
the product of 2 numbers
half of x
twice a number
Division
the quotient of a number and 11
4 divided by a number
4 divided into a number
Equal
Ex
Translate to an equation: Six less than twice the product of 4 and a number is 7.
2
1.2 Commutative, Associative & Distributive Laws
First 2 properties work ONLY for addition & multiplication
1. Commutative - you can change order without changing value
a. Add
b. Mult.
2. Associative – you can change grouping without changing value
a. Add
b. Mult.
3. Distributive:
4. Identity – element that "does nothing"
a. Add (1.5)
b. Mult. (1.3)
5. Inverse – element that "undoes" an operation (gets back to identity)
a. Add
b. Mult.
Note: "Nothing left" in multiplication is NOT zero.
Examples (problem solving)
3
Factoring (reverse distributive law) – write as a product
Ex
Write as a product
Important distinctions:
Term - an added/subtracted piece. So the expression 3x + 2 has _______ terms.
Factor - a multiplied piece. So the expression 4ac has _______ factors.
Ex
List the terms of x2 + 5x + (-7)
Ex
List the factors of 4x(y – 2)
Note:
4
1.3 Fraction Notation
prime number - a number that can't be "broken down" to smaller factors
e.g.
prime factors - the smallest broken down pieces
e.g.
lowest terms – a fraction with no common factors in numerator & denominator
e.g.
Multiplying – smart way: cancel common factors first, then multiply nums. & denoms.
Ex a
Reciprocal (mult. Inverse) – flip fraction
Dividing – invert second fraction and multiply (keep, change, flip)
Ex b
Adding and Subtracting Fractions – must have same denominator
Ex
If denominators differ, must make a common denominator (LCD)
 If the denominators have no common factors, the LCD is the product of the
3
2
and
denominators's, e.g.
LCD:
8
5
 If one number is a perfect multiple of all other denominators, that largest number is the
1 5
3
,
and
denominator, e.g.
LCD:
2 16
4
 Otherwise, listing (or “eyeballing”) multiples of both denoms. can give the LCD, e.g.
7
11
and
12
20
 Prime factorization method of finding LCD’s need for denoms. with many factors
o Find prime factors
o Write all factors of one denominator
o Find missing factors of other denominators (choose highest powers)
5
1.4 Positive and Negative Real Numbers
Number systems you should know:
Whole Numbers = {0, 1, 2, 3…}
Integers = {…-3, -2, -1, 0, 1, 2, 3…}
Rational Numbers = {x | x is the quotient of 2 integers, a/b, b  0}
Irrational Numbers = {x | x is real, but not rational}
Real Numbers = {x | x is a point on the number line}
The Number Line
.…
….
Absolute Value -- units from a number to 0 (always positive)
Ex
6
1.5/1.6 Adding & Subtracting Real Numbers
Adding
1. When signs are same, add amounts, use the same sign as the numbers
2. When signs are opposite, subtract amounts, use sign of the larger (dominant) number.
Subtracting – change subtract to add AND change sign of second number
Additive Inverse (also called opposite) –
Ex
1.7 Multiplication and Division of Real Numbers
 Multiply/Divide by an even number of negatives 

Multiply/Divide by an odd number of negatives 

Multiply by 0

Divide by 0

Multiply by -1:
7
1.8 Exponential Notation and Order of Operations
34 =
Caution:
(-3)4 =
-34 =
Order of Operations
?
?
1.
2.
3.
4.
Parentheses/Grouping Symbols (including horizontal division bars)
Exponents
Multiplication & Division (equal priority, in order, from left to right)
Addition & Subtraction (equal priority, in order, from left to right)
Ex a
Distributive Law and Additive Inverse (Opposite) of Sums
8
2.1 Solving Equations
Solve - get a number that makes the equation true
Solution - a number that makes the equation true
Checking to see if a number is a solution - Procedure
1. Replace the variable(s) with the number
2. Simplify and determine if the equation is true
Goal in Solving:
Anything standing in the way of goal is "junk". How can we get rid of "junk?"
Addition Property of Equality
Equivalent equations  have same solution
A = B and A + C = B + C are equivalent equations
A = B and A - C = B - C are equivalent equations
To "undo" an operation, do the reverse
Examples
Multiplication Property of Equality
A = B and AC = BC have the same solution
A = B and A/C = B/C have the same solution
To "undo" junk, do the reverse
Examples
9
2.2 Using the Principles Together
What if you have both added/subtracted AND multiplied divided junk together?
Playing dumb:
We should get rid of _____________________ junk before __________________junk.
Solving Procedure (summary)
1. Clear equation of parentheses, combine like terms
2. If variables on 2 sides, get rid of 1 term to have variable on one side
3. Get rid of added/subtracted "junk"
4. Get rid of multiplied/divided "junk." You should now have an isolated variable sol.
5. Check solution in original equation
Combining Like Terms
 Same Side

Opposite sides
3 possible outcomes:
1. One solution (conditional equation)
2. No solution (contradiction)
3. Infinitely many solutions (identity)
10
Clearing Fractions
1. Find the LCD
2. Multiply EVERY term by LCD and cancel to eliminate denominators
3. Solve the equation (which now should have no fractions)
Clearing Decimals – book's method (not my preference):
1. Get rid of parentheses, combine like terms.
2. Decide how many places to move (the maximum number of places)
3. Multiply both sides by the power of 10
4. Move the decimal point the same number of places in EVERY term
5. Solve
Clearing Decimals Alternate Method (my preference)
1. Get rid of parentheses, combine like terms.
2. Keep decimals until the end
3. Solve as before
11
2.3 Formulas
Evaluating Formulas
Ex
Solving for a Variable – number value
Ex Find width of a mobile home if the area is 1000 square feet and the length is 40 ft.
Solving for a Specified Variable
Sometimes we need to express a variable in terms of other variables, not numbers, because
we want to do multiple calculations.
For example, suppose the formula for a person's salary is:
S = 200 + 100x, where s is salary, $200 is "flat rate" salary, x is # of computers sold, and
a commission of $100 is received for each computer sold.
When is it desirable to have S isolated (as shown)?
When is it desirable to have x isolated?
Procedure
1. Clear fractions (if needed)
2. Isolate the desired variable by getting rid of “junk”
12
2.4 Applications With Percent
Percent – out of 100. So 57% =
=
Converting percent to decimal: move point 2 places and remove %, or use /100 to convert.
Ex
Convert to decimal:
Converting decimal to percent: multiply by 100%
Solving Percent Equations:
of  multiply
out of  divide
is  equals
percent of whole is part
Example type
unknown part (amount)
OR
part is percent of whole
Word description
Equation
unknown whole (base)
unknown percent
Note: When solving or performing calculations, represent percent as a decimal
13
Ex
A local high school had 550 graduates and 48% attended MJC. How many
graduates attended MJC?
Ex
A 30% discount on shoes saves $15. What is the original price?
Ex
Ed missed 6 questions on a test. If he earned 85%, how many questions were on
the test?
14
2.5 Problem Solving – 2 related unknown quantities
Procedure
1. Familiarize yourself with the problem
2. Translate to algebra
 Choose a variable to represent the unknown (desired) quantity
 Write math expressions using the variable to represent other quantities
 Write an equation
3. Solve the equation
4. Answer the question
5. Check (optional)
Ex a
Common constructions (algebraic expressions) for 2 types
1. “one side is 4 ft more than the other side”
2. “there are twice as many children as adults”
3. “there are 10 coins total”
4. Consecutive integers //
Consecutive even integers // Consecutive odd integers
Ex
A rectangular field has a length which is 20 ft. less than twice the width. If the
perimeter is 200 ft, find the length and width.
Ex
The sum of 2 consecutive street address numbers is 259. Find the house numbers
15
Angle Facts
1. 3 angles of a triangle add to
2. Complementary angles add to
3. Supplementary angles add to
4. Vertical angles are
Ex
In Triangle ABC, Angle A is 141 degrees more than Angle B. Angle C = Angle B. Find
the angles.
Ex
Find the angle whose supplement is 30 degrees less than 4 times its complement.
16
2.6 Solving Inequalities
Algebraic Form
Set Builder
Notation
Graph Form
x>3
__________________________
x < -2
__________________________
80 < x < 90
__________________________
Interval
Notation
Solving Inequalities – Goal: Isolate x
Tools for solving:
Addition Property of Inequalities - (similar to equality)
A < B and A + C < B + C have the same solution
A > B and A + C > B + C have the same solution
(The rules also hold true for subtracting C from both sides, and also for > or <)
Ex a
Multiplication Property of Inequalities - tricky – it's different from equality in some cases
Rule 1 Multiply or divide by a positive number - keep direction the same
For C > 0, A < B and AC < BC have the same solution
To show:
Rule 2 Multiply or divide by a negative number - change direction
For C < 0, A < B and AC > BC have the same solution
To show:
(The rules also hold true for dividing by C on both sides, and also for >, <, or >)
Note: If you exchange sides, you must exchange the direction of the inequality, unlike
equations which are the same on both sides
x =3  3=x
x < 5  5 > x (arrow always "points to" the smaller quantity)
17
2.7 Applications With Inequalities
Phrase
gifts for $10 and under
kids under 3 eat free
the minimum payment is $50
kids over 80 lb. cannot ride
Variable
x = cost of gift
x = age of kid eating free
x = amount I must pay
x = weight of kid who can’t ride
Inequality
Ex Al can spend no more than $1000 on housing and utilities. If rent is $725, water and
gas are $135, how much can be spent on electricity?
Ex Mel wants an “A” average on tests. What does he need on the last test if his first 4
scores are 82, 91, 88, and 95?
Ex Phone plan A charges $40 for 0 – 500 minutes and $0.50/extra minute. Plan B charges
$0.25/minute. Let x = number of minutes used.
1) If x < 500 minutes, when is Plan A better?
2) If x > 500 minutes, when is Plan A better?
18
3.1 Reading Graphs; Plotting Points, Scaling Graphs
Graphs – often used to connect 2 quantities in a relationship
Bar Graphs – good for comparing categories
Which increase in education causes the greatest
increase in income?
Salary (thousands)
Salary vs. Education (2003)
60
50
40
How does the salary of an AA holder compare
with someone with no diploma?
30
20
10
0
No
diploma
HS
AA
BA
MA
If someone wanted a salary of $40,000 or higher,
what educational level is recommended?
Degree
Line Graphs – good for showing a progression over time
Week
0
10
20
Weight
150
146
154
30
164
40
182
How much does this person weigh at Week 25?
Weight During Pregnancy
Weight in Pounds
185
180
175
170
165
During which 10-week period is the weight gain the
greatest?
160
155
150
145
140
0
10
20
30
40
Week
Scatter Diagram – good for plotting points when relationship is not known
# of Absences 3
8
0
12
4
9
Final score
80
72
95
52
85
68
5
80
What is the relationship between
absences and final grades?
Absences and Final Grades
100
Final Score
90
80
70
If a student has 6 absences, what would
his/her final grade be expected to be?
60
50
40
0
2
4
6
8
Number of Absences
10
12
14
19
Pie Graphs – good for showing parts of a whole
Monthly Expenses
If annual income is $50,000, how much is spent
on housing?
How does the cost of transportation compare to
food?
The Rectangular (Cartesian) Coordinate System
axis – number line used to locate a point
point – a location in space
ordered pair (x,y) – the coordinates of a point (x is always first)
quadrants - 4 regions defined by the x and y axes
origin Ex Plot the points A(3,0), B (-2, 4), C (0, -2), D(-3, -1)and tell what quadrants they’re in
20
3.2 Graphing Linear Equations
A solution gives a true equation when the variables are replaced with numbers.
Ex a Test if (1, 3) is a solution of 2x + 3y = 6
Ex b Test if (3, 0) is a solution of 2x + 3y = 6
Graphing these points:
By inspecting the graph, is (0,0) a solution of 2x + 3y = 6?
Observe
1.
2.
3.
Completing an Ordered Pair (finding a solution)
Ex
Graphing a Linear Equation
1. Select one value of a pair (x or y), and calculate the other value to get a solution
2. Find at least 1 more solution
3. Plot the points and connect the dots
21
Ex
Complete a table of values and draw the graph of
x
y
Linear Equations (in 2 variables) have 2 common formats:
1. y = mx + b 
2. Ax + By = C 
Some people prefer to convert all equations to slope-intercept form for graphing (optional).
Ex
Ex
Convert
Graph the equation
22
3.3 Graphing and Intercepts
Intercepts - special points (MUST be an ordered pair)
x-intercept: the x value where the line crosses the x-axis (y = 0)
y-intercept: the y value where the line crosses the y-axis (x = 0)
To find each intercept, set the opposite coordinate = 0
Ex a Find the intercepts and graph of
Special Cases
1. Horizontal Line
2. Vertical Line
3. Line Thru the Origin
Summary: For the linear equation Ax + By = C
1. A = 0  no x term, y only 
2. B = 0  no y term, x only 
3. C = 0  no constant (number term) 
23
3.4 Rates
rate – a ratio of unlike units. Purpose: to show the relationship of 2 different quantities
key word: per –
Ex a A car is rented for 3 days and travels 1500 miles . The rental cost is $450, and the
10-gallon tank is filled 4 times. The total cost of gas is $152.
1) Find the gas mileage in
2) Find the total cost/mile
3) Final the rental cost/day
4) Find the rate of travel in miles/day
Ex
Tuition at a college in 2007 was $5000. In 2013, tuition was $8000.
1) Find 2 ordered pairs that represent this data.
2) Calculate the rate of increase
3) Graph the data
Ex A cable plan starts at $60/month, with $5 extra for pay per view (PPV) movies.
1) Write an equation for the monthly bill, with x = number of PPV movies watched
2) Calculate the cost for 1, 10, and 20 PPV movies watched in a month
24
3.5 Slope
slope – the slant or steepness of a line
y -y
rise
change in y (vertical)
slope = m =
=
 2 1
run change in x (horizonta l) x 2  x1
y -y
m  2 1 for 2 points on a line with coordinates (x1, y1) and (x2,y2)
x 2  x1
Ex
Vertical and Horizontal Lines
Ex
Observations:
25
3.6 Slope- Intercept Form
3 Common Forms of Equations
1. General Form – best for “nice” presentation
2. Slope-Intercept Form – best for graphing
3. Point-Slope Form – most flexible for getting an equation from word information
Note: The 2 acceptable forms for final answers are general and slope-intercept forms
Finding Slope and Intercept from an equation
Ex
What if equations are not in slope-intercept form?
1. Isolate y
2. Write x term first, plain number second
Graphing, using Slope-Intercept form (y = mx + b)
1. Plot the y-intercept (anchor)
2. Use the slope to count up/down and left/right to find another point
3. Draw a line connecting the dots
Occasionally, m and b are given directly
26
Parallel and Perpendicular Lines
2 lines are parallel if they have
2 lines are perpendicular if their slopes are:
a)
b)
Note: To find slope, get equation in y = mx + b form. The coefficient of x is m, the slope.
Slope includes ONLY the number in front of x, not x itself.
Ex Are the lines y = 3x + 4 and x + 3y = 1 parallel, perpendicular, or neither.
27
3.7 Point-Slope Form
Finding an Equation from Slope and 1 point
Note: Slope-Intercept is preferred final form:
Finding an Equation from 2 points
Ex
Find the equation of the line passing through the points (-4, -1) and (2, 3)
Ex
(-2, 1)
Find the equation of the line perpendicular to y = ½ x + 7, passing through the point
28
9.4 Inequalities in 2 Variables
Graphing 1 linear inequality - Procedure
1. Graph the inequality as if it were an equation
a. Use solid line for > or <
b. Use dotted line for > or <
2. Decide where to shade using a test point (alternate method: isolate y on left)
Ex
Sal has $1200 monthly to spend on rent and utilities.
a) Let x = rent cost and y = utility cost. Write an inequality expressing the possible rent
and utility costs.
b) Graph the inequality
c) Give some examples of ordered pairs that satisfy the inequality
Solving (Graphing) a System of 2 Linear Inequalities - Procedure
1. Graph the 1st line and shade its solution
2. Graph the 2nd line and shade its solution
3. The overlapping region is the final solution
Note: The double shaded quarter is opposite the "empty" quarter.
29
8.1 Solving Systems of Equations -- Graphing Method
A system of equations has 2 (or more) equations with 2 variables (or more
We are interested in finding points that are solutions of both equations (the system solution)
Ex a Is
a solution of the system:
3 possible outcomes (3 kinds of systems)
1. The 2 lines intersect at one point
y = 3x
y=-x+4
-
2. The 2 lines never intersect
y=x–1
y=x+3
-
3. The 2 lines lie on top of each other
y=x
3x – 3y = 0
-
The most common (and expected) outcome is one solution.
30
Translating Word Problems to a System of 2 Equations
Ex
Translate (don’t solve)
There are 3 more than twice as many children as adults at a park. If there are 21 people,
how many children and how many adults are there?
Ex
Translate (don’t solve)
Cashews cost $7/lb and walnuts cost $4/lb. If 18 lb. of a mix costing $5/lb is made, how
many lb. of each type of nut is used?
Ex
Translate (don’t solve)
A soup kitchen bought 5 turkeys and 6 hams for $150. Salvation Army bought 6 turkeys and
4 hams at the same prices for $132. What are the ham and turkey prices?
31
8.2 Solving Systems – Substitution or Elimination
Graphing gives a "big picture" view of the solution, but it has limitations:
Substitution and Elimination are more precise
Procedure - Solving by Substitution
1. Isolate 1 variable in 1 equation
2. Substitute that variable into the other equation. You should now have an equation with
variable.
3. Solve that equation for the variable
4. Solve for the other variable. Write your solution as an ordered pair
5. Check both equations (optional)
32
Suppose we have a system where one variable is not easily isolated:
4x + 3y = 30
2x – 3y = 6
Substitution gives messy fractions (ugh!)
Addition Property of Equations
If a = b and c = d, then a + c = b + d
Procedure for Solving by Elimination
1. Write both equations in general form (Ax + By = C) and stack with like types in a column.
2. Choose which variable to eliminate
3. Multiply one or both equations so the coefficients have opposite sign same amount
4. Add 2 equations. One variable should drop out
5. Solve for the remaining variable
6. Solve for the other variable. Write as an ordered pair
7. Check solution in both equations (optional)
33
8.3 Solving Applications – Systems of 2 Equations
Procedure for solving
1. Choose variables to represent the desired quantities (choose sensible ones!)
2. Write a system of equations (usually 2) from the information given
3. Solve the system
4. Answer the question
5. Check
Money Value
Ex
Zoe bought burgers for $3 and hot dogs for $2. If 20 items were bought for $47, how
many of each type was bought?
Interest
Ex
An investor splits $10,000 between a “safe” fund at 5% and a “risky” fund at 20%. If
$1100 in interest was earned, how much was in each account?
Mixture
Ex
If 1% and 4% milk are mixed to make 24 oz. of 2% milk, how much of each type of
milk is used?
34
Distance/Rate/Time
Wind/Current
Let
s = speed of plane in still air (the plane speed without wind)
w = wind speed
s + w = speed with the wind
s –w = speed against the wind
Ex
A plane flies 200 mph with the wind and 140 mph against the wind. Find the plane
speed (without wind) and the wind speed.
35
4.1 Exponents (Positive) and Their Properties
Rules
1. Product: a m  a n  a mn
2. Quotient:
am
 a m-n
n
a
3. Zero as exponent: a 0  1
4. Power to a power: (a m )n  a mn
5. Product to a power: (a  b)m  am  bm
m
am
a
6. Quotient to a power:    m
b
b
7. One as exponent: a1 = a
Ex a
36
4.2 Negative Exponents and Scientific Notation
1
1. Negative exponent: a -n  n
a
2. Negative exponent fraction:
a -n b m

b m a n
37
Scientific Notation - useful for very large and very small numbers
Form: N.dd…d X 10m where N a non-zero digit, d is a digit, m is the exponent
 must have exactly one non-zero digit left of the decimal point
 any number of digits (zero or more) is allowed to the right of the decimal point
Ex a Which of the following are in scientific notation?
Procedure – Converting Place Value Form to Scientific Notation
1. Write the decimal point immediately after the first non-zero digit
2. Count how many places you moved to get there
3. Write that number as the exponent
4. Write the sign of the exponent: positive for large numbers (> 1)
negative for small numbers (< 1)
Procedure – Converting Scientific Notation to Place Value Form
1. Move the decimal point the number of places in the exponent
 Make bigger (move to right) for positive exponent
 Make smaller (move to left) for negative exponent
2. Fill in zeroes where necessary
Multiplying and Dividing with Scientific Notation
Scientific notation allows you to round very large (or small) numbers so that estimates for
calculations can be made.
Ex
The federal debt in 2012 was $16,160,000,000,000, and the U.S. population was
312,800,000.
a) Write these 2 numbers in scientific notation:
b) Round these numbers to one significant digit, keeping powers of 10.
c) Divide the debt by the population to find out how much money is owed by each person.
38
4.3 Polynomials
Term = one element (with no addition or subtraction separating pieces)
Monomial – type of term: a number, variable, or product of numbers and variables
Examples:
Counterexamples (not monomials):
Note: A monomial may have a number in the denominator, but not a variable in denom.
binomial:
terms added or subtracted; e.g.
trinomial:
terms added or subtracted; e.g.
polynomial: one or more terms added or subtracted; e.g.
degree of a term – the number of variable factors (if one variable, the degree is the exponent)
Ex
Fill in the chart for the each of the terms of the polynomial:
Term
Degree of term
Coefficient
Variable Part
Deg. of Polynomial
descending order – polynomials with different exponents are written with highest powers first
leading term – the first term of a descending order polynomial; the highest power term
degree (or order) of a polynomial - the degree of the highest power term
Ex Write the polynomial in descending order, then find the leading term and degree of the
polynomial:
like terms – have exactly the same variable part, but may have different numbers
unlike terms – have different variable parts
Ex



Are the following pairs like or unlike?
39
Combining Like Terms – put numbers together, keep same variable part
Ex
Evaluating a Polynomial – plug in a number for each variable
Ex
The total number of games played by a sports league is described the formula:
n2  n
N =
2
Ex
The height of a falling object is described the formula
h = ho – 16t2
How far does a rock fall in 6 seconds?
40
4.4 Addition & Subtraction of Polynomials
Adding – Horizontal Method (good for problems with missing terms)
Ex
Adding – Vertical Method (stack up like terms)
Ex
Subtracting - take opposite (additive inverse) of every term in 2nd polynomial
Ex
Ex
Find a polynomial for the sum of the areas of the 4 small rectangles:
41
4.5 Multiplying Polynomials
Multiplying monomials – gather & multiply coefficients, then gather & multiply each variable
Ex
Multiplying a monomial and a polynomial – use distributive law
Ex
Multiplying 2 polynomials:
Horizontal Method (repeated distributive law)
Ex b
Vertical Method (like arithmetic multiplication)
Ex c
Recall:
X
Box Method
Ex d
421
352
42
4.6 Special Products
Multiplying 2 binomials
FOIL - First, Outer, Inner, Last
F  F  O  O  I  I  L  L (4 terms)
Formulas to memorize:
1. (A + B)(A – B) = A2 – B2
2. (A + B)2 = A2 + 2AB + B2
3. (A – B)2 = A2 – 2AB + B2
To show #1:
To show #2:
Common mistakes:
(A + B)2  A2 + B2
So (x + 3)2  x2 + 9
2
2
2
(A - B)  A - B
So (x - 1)2  x2 -1
Note: There is no formula that makes A2 + B2
43
4.7 Polynomials in Several Variables
Degree of a term is the product of the variable factors. For 2 or more variables 
Ex Find the coefficient and degree of each term, and the degree of the polynomial
– 2x + x7y2 - 5xy2 – 6y4 + 3
Term
Coefficient
Variable Part
Degree of
Term
Like terms must have exactly the same ___________________
Add/Subtract Polynomials
Multiply Polynomials
Degree of
Polynomial
44
4.8 Division of Polynomials
Cancellation (monomials):
Dividing Polynomials by Monomials
abc d a b c d
   
Rule:
(Rule also applies for subtraction)
x
x x x x
Dividing Polynomials by Binomials – use long division
Recall arithmetic:
1 2 387
quotient
divisor dividend
Procedure – Long Division
1. Divide 2 leading terms. Put in quotient above correct term.
2. Multiply quotient piece by divisor
3. Subtract
4. Repeat until terms are used up
5. Write remainder over divisor.
45
5.1 Intro to Factoring
Factors - pieces multiplied to make a product
It is sometimes desirable to write things as factors (pieces)
Ex
Write 12 as the product of 2 factors in as many ways as possible:
Ex
Write 3 factorizations of 15x3
Factoring Out a Common Factor - Reverse of distributive law
Multiplying
Factoring
When possible, factor out the Greatest Common Factor (GCF). We usually do this by
“eyeballing”, and taking out more factors if they remain.
Factoring out the GCF – the reverse of the distributive law
1. Find the GCF (by eyeballing or other method) and write it in front
2. Divide the GCF out of each term
3. Write the "leftovers" inside parentheses
46
Finding the GCF (Complete/Long Method)
1. Write all numbers in prime factored form with exponents
2. For each base, choose the smallest exponent common to all terms
3. Find the product.
Factoring by Grouping – take out identical “clumps” in parentheses
47
5.2 Factoring Trinomials (of type 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
2
(x + 1)(x + 2) = x + x + 2x + 2 =
x2 + 3x + 2
2
(x – 3)(x – 5) = x – 5x – 3x + 15 =
x2 – 8x + 15
(x – 4)(x + 3) = x2 + 3x – 4x – 12 =
x2 – x – 12
2
(x – 2)(x + 5) = x + 5x – 2x – 10 =
x2 + 3x – 10
Observations on signs
1. If c is positive 
a. If b is positive 
b. If b is negative 
2. If c is negative 
a. If b is positive 
b. If b is negative 
Observations on numbers
1. c is the product of the 2 binomial numbers 
2. If the 2 binomial numbers have the same sign, b
3. If the 2 binomial numbers have different signs, b
48
5.3 More Factoring Trinomials (ax2 + bx + c)
ax2 + bx + c factors to (?x + ?)(?x + ?)
Factoring by FOIL (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 5.2
 Rules for numbers require "mix and match" to get the middle number, b
Factoring by Grouping – long process, no guesswork
1. Multiply a  c
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
49
5.4 Factoring - Perfect Square Trinomials & Difference of Squares
x
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15
2
x
1
4
9
16 25 36 49 64 81 100 121 144 169 196 225
Special formulas (same as before, written in reverse):
1. A2 – B2 = (A + B)(A – B)
2. A2 + 2AB + B2 = (A + B)2
3. A2 – 2AB + B2 = (A – B)2
Ask: Does my binomial or trinomial fit one of these 3 patterns
Binomials – Use 1st formula for binomials, last 2 formulas for trinomials
Trinomials - Use formulas 2 or 3
Factoring Completely
50
5.5 Factoring Sums and Difference of Cubes
x
1
2
3
4
5
6
10 1/2 0.1
3
x
1
8
27 64 125 216
1. A3 + B3 = (A + B)(A2 – AB + B2)
2. A3 – B3 = (A – B)(A2 + AB + B2)
Common Errors:
A3 + B3  (A + B)3 So (x+2)3  x3 + 23
A3 – B3  (A – B)3
Note: A2 – B2 can be factored 
A2 + B2
51
5.6 Factoring Summary
Factoring Summary
1. Can a GCF be factored out? (5.1)
2. 2 terms: Is it (A2 – B2) or (A3 + B3) or (A3 – B3)? (5.4, 5.5)
3. 3 terms: Is it (A2 + 2AB + B2) or (A2 – 2AB + B2)? (5.4)
Does trinomial  2 binomial factoring work easily? (5.2)
Is there a number in front of x2 for trinomial  2 binomials? (5.3)
4. 4 terms: Does factoring by grouping work?
5. Can any factors be further factored?
52
5.7 Solving Quadratic Equations – Factoring
Quadratic equation – 2nd degree
Standard form: ax2 + bx + c = 0
Zero Product Principle – basis for solving quadratic equations
ab = 0 if and only if a = 0 or b = 0 (one or both 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.
What is the maximum number of solutions of a quadratic equation?
How many solutions could a cubic equation potentially have?
53
5.8 Solving Applications of Quadratic Equations
Numbers
Ex
The product of 2 consecutive integers if 4 less than 4 times their sum. Find the
numbers.
Rectangles: A = LW; P = 2L + 2W
Ex
A garden is 11 ft. longer than it is wide. The total area is 60 sq. ft. Find the
dimensions.
Triangles:
Pythagorean Theorem: In a right triangle with hypotenuse c and legs a & b,
a 2 + b 2 = 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.
Given Formulas (height, time)
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)
b)
c)
d)
e)
What is the height of the ball after 2 seconds?
For what values of t is h = 0?
Interpret the values from b) in real life terms.
For what values of t is h = 240 ft?
After how many seconds do you think the ball will reach its peak?
54
7.1 Introduction to Functions
A correspondence or relation is a system that connects 2 sets of quantities (e.g., x and y) to
each other.
Domain – set of all possible x values (inputs)
Range – set of all possible y values (outputs)
Some Examples
1. A set of ordered pairs (5 children)
Age
4
7
9
12
9
Weight 42
61
75
92
68
The ordered pairs are (4, 42), (7, 61), (9, 75), (12, 92), (9, 68)
2. A vending machine
A 
B 
C 
D 
E 
3. An equation with x and y
a) y = 3x – 1
b) y = x2
4. A graph with x and y
Function – a correspondence where each input (x) has exactly one output (y)
 never 2 or more outputs for same input
 OK to have 2 inputs produce same output
 a function is predictable
Deciding if ordered pairs are functions
1. Check to see if 2 or more pairs have same inputs
2. If different outputs, not a function
55
Deciding if a graph is a function
Vertical Line Test - A graph is not a function if any vertical line cuts the graph at more than
one point
Ex
Which of the following are functions?
Deciding if an equation is a function
Odd Powers Test – If an equation contains one y term:
1. If the exponent on y is odd (e.g. y, y3, y5…) it is a function
2. If the exponent on y is even (e.g. y2, y4, y6…) it is not a function
Function Notation
f(x) – spoken as “f of x”  NOT multiplication
We often replace y with f(x).
We can also use other symbols in function notation, which are connected to quantities
e.g. C(t), where C is cost, t is time in minutes spent on a phone
56
6.1 Rational Expressions (Canceling Fractions)
Rational Expression: Can be written as P/Q, where P and Q are polynomials , Q  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
x3
undefined?
(x  1)(x  5)
Simplifying (Cancelling)
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)
PK P
where K  0

QK Q
57
6.2 Multiplication and Division of Rational Expressions
P R PR
Multiplying:
 
Q S QS
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:
P R P S PS
   
Q S Q R QR
58
6.3 Adding and Subtracting; Least Common Denominator (LCD)
P R PR
P R P R
and
for Q  0
 
 
Q Q
Q
Q Q
Q
Caution: 2 fractions must have same denominator. If not, a common denominator (LCD)
must be used.
Rules:
Procedure -- Adding/Subtracting with same denominator
1. Add or subtract numerators, keep denominator
2. Factor and/or cancel if possible
Ex
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
Building Fractions to their LCD
1. Find the LCD
2. Compare the old denominator with the LCD to find “what’s missing”
3. Multiply top and bottom by the missing factor
59
6.4 Addition & Subtraction with Unlike Denominators
Procedure - Add/Subtract with unlike denominators
1. Find the LCD
2. Build each fraction to match LCD
3. Add or subtract as above
Recall: The LCD is the least common multiple of the denominators. It is as large as or larger
than each of the individual denominators.
60
6.5 Complex Fractions/Rational Expressions
 Typically have 4 layers (2 top, 2 bottom)
 We can treat 4 layers as 2 separate fractions, divided
2 Solving methods
Method 1: Divide by 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
61
6.6 Solving Rational Equations
Recall:
Expressions
 no equal sign
Equations
 has equal sign

Goal: Keep denominator by putting
numerator over LCD

Goal: get rid of denominator by
multiply by LCD

Final answer may have mixed
variables and numbers

Final answer: x = number (variable is
isolated)

Process is to

Process is to
Solving Procedure
1. List the bad points
2. Find LCD and 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
62
6.7 Applications (these are in reverse order from the book)
Proportions
proportion - 2 ratios that equal each other
e.g.
a c
 then ad = bc (changes fractional equations to nonfractional)
b d
Solving Proportion Equations
Ex b
cross multiplying: If
Shortcut cancellations:
1. You can cancel straight across (cancel 2 numerators OR cancel 2 denominators)
2. You can cancel straight up & down (a numerator with a denominator)
3. You CANNOT diagonally cancel
Caution:
1) Don't confuse
20
9
20 9


with
3
50
3 50
2) You can't cancel when terms are connected by addition
Word Problem Chart
Quantity 1
Case 1
Case 2
Quantity 2
Case 1
OR
Quantity 1
Quantity 2
Case 2
63
Distance/Rate/Time
d = rt; isolating for different variables, r = d/t or t = d/r
Ex
A plane flies 330 miles with the wind in the same time it takes to fly 270 miles agains
the wind. The plane speed is 200 mph. Find the wind speed.
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)
Ex
Jay takes 12 hrs. to build a fence alone, and Bo takes 16 hrs. How long would it take
them working together?
Ex
Working together, Julie and I take 10 minutes to wash a load of dishes. Alone, it
takes me 15 minutes. How long does it take Julie alone?
64
10.1 Radical Expressions and Functions
square rooting - the reverse process of squaring
Suppose we are given x2 = 25. What is the "unsquare" of 25?
“Generic” Square Root: “c” is a square root (one of 2 possible) of “a” if c2 = a
Principal Square Root: “c” is the only principal (non-negative) square root of "a" c =
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
Cube Roots
If c3 = a, the signs of “c” and “a” are the same, so the “generic” and “principal” cube roots are
the same (
3
8
"The cube root of 8"
3
8
“The cube root of – 8”
65
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
Higher roots of negative numbers
The nth root of a negative number n  a is

if n is odd

if n is even
Roots of Expressions with Variables
To be a perfect square, a variable exponent should be:
To be a perfect cube, a variable exponent should be:
Absolute Values of Variables in Roots
Function values
66
10.2 Rational Numbers (Fractions) as Exponents
By definition:
a ½= a
a 1/3 = 3 a
a 1/n = n a
To verify:
Other Fractional Exponents
 
m
a m/n = n a m  n a
a m/n = (am)1/n = (a 1/n)m
Negative Fractional Exponents - as before, neg. exponent means
Laws of Exponents (same as before)
Recall:
Product rule: a m  a n  a mn
am
Quotient rule: n  a m-n
a
Power rule: (a m )n  a mn
Product to a power rule: (ab)m = ambm
67
10.3 Multiplying Radical Expressions
Multiplying: Product Rule (also applies for higher roots)
a b  ab and ab  a b for any real numbers “a” and “b”
Caution: The Product Rule does NOT work for addition
Simplifying: A radical is completely simplified when no perfect square factor is inside the
radical symbol (or perfect nth power factor for higher roots)
2 Methods for simplifying:
 Method 1 – Find prime factorization & take out even powers (or multiples of nth
powers for nth roots)
 Method 2 – "Eyeball" perfect squares that are easy to see & take them out until no
perfect squares are left (or perfect nth roots)
Radicals with Variables
Higher roots/Powers
68
10.4 Dividing Radical Expressions
Dividing: Quotient Rule
a
a
a
a
and
for real numbers “a” and “b”,


b
b
b
b
b  0
Rationalizing the Denominator - Requirements
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 - one term (monomial)
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
69
10.5 Combining Several Radical Terms
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
Additing & Subtracting:
Don't confuse with multiplication:
ab  a  b
Radicals are similar to variable like terms – only like types can be added/subtracted
Multiplying – distributive law
Multiplying – FOIL
70
Rationalizing Denominators of Binomials (2 terms)
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
71
10.6 Solving Radical Equations
Principle of Powers: If a = b, then an = bn for any exponent n.
Goal: Isolate the variable
Recall: Squaring is the reverse of square root, so squaring a radical "undoes" it
Solving 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!
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"
72
10.8 Complex Numbers
The number i is defined as i =
i2 = -1
- 1 (a non-real number)
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
73
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
74
11.1 Equations – Square Root & Complete the Square
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 real solutions  x2 = positive number (or (stuff)2 = positive #)
2. no real solution  x2 = negative number
3. one real solution  x2 = 0
75
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
2
3.
4.
5.
6.
b
nn =  
2
Add it to both sides.
Factor the perfect square. Write it as a square that looks like (x + b/2)2
Square root both sides
Solve for x
76
11.2 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:
 b  b 2  4ac
 b  b 2  4ac
or x =
2a
2a
Putting the 2 together:
x=
 b  b 2  4ac
2a
Your book derives this formula. We won't!
x1,2 =
77
Compound Interest
A = P(1 + r)t
Application of Quadratic Formula and Fractional Equations
Based on the compound interest formula A = P(1 + r)t, if you solve for P, the equation looks
like:
A
P=
(1  r) t
Problem:
Gloria wants to get a BA degree in 3 years. After paying for her first year up front, she has
$11,000 left to pay for the last 2 years. Each year costs $6000, and payment is due at the
beginning of each year. At what interest rate must she invest to earn enough interest to
cover the costs? The formula for investments at 2 annual payments, with interest
compounded annually is:
A
A

P=
; P = principal invested, A = amount paid out, r = interest rate
1  r (1  r) 2
Solution:
From our data, P = 11,000 and A = 6000. Plugging into the formula:
11,000 =
6000 6000

1  r (1  r) 2
To get rid of fractions, use the LCD: (1 + r)2
(11000)(1 + r)2 = 6000(1 + r) + 6000
(11000)(1 + 2r + r2) = 6000 + 6000r + 6000
11000 + 22000r + 11000r2 = 12000 + 6000r
11000r2 + 16000 r – 1000 = 0
1000(11r2 + 16r – 1) = 0
 16  16 2  4(11)( 1)
By the quadratic formula, r =
= 0.06 or -1.5
2(11)
So the interest rate needed is .06 = 6%.
Business majors' note:
When calculating interest we commonly ask the question, "If I invest P dollars for t years at
r% interest, how much will I have at the end?"
But sometimes the question needs to be asked in reverse. For example, "If I want to have an
income of A dollars each year paid out over t years, how much Principal do I have to invest
originally at r% interest?" This kind of calculation is called Net Present Value (NPV).
78
11.3 Studying Solutions of the Quadratic Equations
Recall: The equation ax2 + bx + c = 0 has solutions:
 b  b 2  4ac
x1,2 =
2a
The inside of the square root is sometimes called the discriminant.
 If b2 – 4ac > 0 
root(s)

If b2 – 4ac = 0 
root(s)

If b2 – 4ac < 0 
root(s)
Ex a For each of the equations, determine the number and type of solutions without solving:
x2 + 4x + 6 = 0
x2 – 6x + 9 = 0
x2 – 4x – 5 = 0
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.
Writing Equations from Solutions
Ex b Write an equation if the solutions are {-5, 2/3}
79
11.4 Applications of Quadratic Equations
Distance/Rate/Time
Ex
I drive to Carlsbad (330 miles). My husband drives 11 mph faster and saves 1 hour.
How fast is each of us driving?
distance
Rate
Time
Pendulum Formula
l
T  2
g
Motion Formulas
v f  vo  at
s  vo t 
1 2
at
2
(Assume acceleration due to gravity is 32 ft/sec.)
Ex
A cliff is 64 ft. high
1) How long does it take a diver to hit water, assuming no initial velocity?
2) How fast is the diver traveling at when entering the water?
80
11.6 Graphing Quadratic Equations - Plotting Points
An equation or function can be graphed by calculating and plotting points. For non-linear
equations, more than 2 points are needed to visualize of the shape of the graph.
Ex a Graph the function f(x) = x2
x f(x)
-2
-1
0
1
2
Ex b Graph the equation y = - x2 + 1
x
y
-2
-1
0
1
2
Compare these 2 parabolas. What do you observe about :
- the direction of opening?
- the intercepts?
Ex c Graph the function f(x) = x2 + 4x + 4. What should be done to “see” the shape?
x
y
-2
-1
0
1
2
81
Section 11.6 Homework Problems
Sketch the graph of each equation:
1. y = x2 – 3
2. y = -x2 + 4
3. y = 2x2 – 1
4. y = - x2 + 2x
5. y = ½ x2 – 4
6. y = -3x2
7. y = - x2 – 2x – 1
8. y = x2 – 4x + 3