Math 70 Course Pack Beginning Algebra

Spring 2015

Instructor: Yolande Petersen

DO NOT BUY THESE NOTES IF YOU HAVE A

DIFFERENT INSTRUCTOR

How to assemble these pages in a notebook (3-ring binder needed): 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 space for handwritten extra notes on the right. To make your notebook look like mine: 1. If there is a staple, remove it. 2. Separate the Exam Reviews (back, right-side up) from the Lecture Notes (upside down) 3. Take the stack of Lecture Notes, and KEEPING ALL PAGES IN A STACK IN THE SAME

ORDER, flip the whole stack upside down, keeping the punched holes on the left. You should now have the blank back of page 1 on top, holes on the left.

4. Insert these pages into the binder. When you turn the first page, page 1 will be on your left. It should look like a book, with the printed page on the left and the blank page for writing extra notes on the right.

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 assemble the pages however you like.

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.

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Chapter 1 (Quick Summary) 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

Examples (more in book) Addition

the sum of a number and 2

5 more than a number

3 added to t

a number increased by 10

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 4 and a number

6 times p

half of x

twice a number

2

Division

the quotient of a number and 11

4 divided by a number

4 divided into a number

the ratio of debt, d, to equity, e

$15 per 5 gallons

Equal Ex Translate to an equation: Six less than twice the product of 4 and a number is 7. Ex Ray made $35 less this week than last week. If last week’s salary was $520, how much did he make this week?

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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)

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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

denominators's, e.g. 5

2 and

8

3 LCD:

If one number is a perfect multiple of all other denominators, that largest number is the

denominator, e.g. 4

3 and

16

5 ,

2

1 LCD:

Otherwise, listing (or “eyeballing”) multiples of both denoms. can give the LCD, e.g.

20

11 and

12

7

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 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} Real Numbers = {x | x is a point on the number line} Irrational Numbers = {x | x is real, but not rational}

Ex a List each number next to the type that applies: , 0.7, 11, -29, 0, 60. ,39 ,3

4

Whole numbers

Integers

rational numbers

irrational numbers

real numbers

The Number Line

.… …. Absolute Value -- Ex

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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:

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1.8 Exponential Notation and Order of Operations 34 = Caution: (-3)4 = -34 = Order of Operations ? ?

1. Parentheses/Grouping Symbols (including horizontal division bars) 2. Exponents 3. Multiplication & Division (equal priority, in order, from left to right) 4. Addition & Subtraction (equal priority, in order, from left to right)

Ex a Distributive Law and Additive Inverse (Opposite) of Sums

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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

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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 first, __________________junk last. 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)

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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. Decide how many places to move (the maximum number of places) 2. Multiply both sides by the power of 10 3. Move the decimal point the same number of places in EVERY term 4. Solve Clearing Decimals Alternate Method (my preference) 1. Keep decimals until the end 2. Solve as before

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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?

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Procedure 1. Clear fractions (if needed) 2. Get all terms with specified variable on one side, all other terms on other side 3. Factor out desired variable (if 2 or more terms) 4. Get rid of “junk”

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5. 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 OR part is percent of whole

Example type Word description Equation

unknown part (amount)

unknown whole (base)

unknown percent

Note: When solving or performing calculations, represent percent as a decimal

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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. Check (optional) 5. Answer the question 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”

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2.6 Solving Inequalities Algebraic Form

Set Builder Notation

Graph Form Interval Notation

x > 3

__________________________

x < -2

__________________________

80 < x < 90

__________________________

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)

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2.7 Applications With Inequalities

Phrase Variable Inequality

gifts for $10 and under x = cost of gift

kids under 3 eat free x = age of kid eating free

the minimum payment is $50 x = amount I must pay

kids over 80 lb. cannot ride x = weight of kid who can’t ride

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?

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3.1 Reading Graphs; Plotting Points, Scaling Graphs Graphs – often used to connect 2 quantities in a relationship Bar Graphs – good for comparing categories

Salary vs. Education (2003)

0

10

20

30

40

50

60

No

diploma

HS AA BA MA

Degree

Sala

ry (

tho

usan

ds)

Which increase in education causes the greatest increase in income? How does the salary of an AA holder compare with someone with no diploma? If someone wanted a salary of $40,000 or higher, what educational level is recommended?

Line Graphs – good for showing a progression over time

Week 0 10 20 30 40

Weight 150 146 154 164 182

Weight During Pregnancy

140

145

150

155

160

165

170

175

180

185

0 10 20 30 40

Week

Weig

ht

in P

ou

nd

s

How much does this person weigh at Week 25? During which 10-week period is the weight gain the greatest?

Scatter Diagram – good for plotting points when relationship is not known

# of Absences 3 8 0 12 4 9 5

Final score 80 72 95 52 85 68 80

Absences and Final Grades

40

50

60

70

80

90

100

0 2 4 6 8 10 12 14

Number of Absences

Fin

al

Sco

re

What is the relationship between absences and final grades? If a student has 6 absences, what would his/her final grade be expected to be?

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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

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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

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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 Convert Ex Graph the equation

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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)

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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 Graphing Ex

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3.5 Slope slope – the slant or steepness of a line

slope = m = run

rise=

12

12

xx

y- y

l)(horizonta x in change

(vertical) y in change

12

12

xx

y- ym

for 2 points on a line with coordinates (x1, y1) and (x2,y2)

Ex Vertical and Horizontal Lines

Ex

Observations:

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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

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Parallel and Perpendicular Lines 2 lines are parallel if they have 2 lines are perpendicular if their slopes are:

a)

b)

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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

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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.

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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

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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

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4.1 Exponents (Positive) and Their Properties Rules

1. Product: nmnm aaa

2. Quotient: nm-

n

m

a a

a

3. Zero as exponent: 1a0

4. Power to a power: mnnm a)(a

5. Product to a power: mmm bab)(a

6. Quotient to a power: m

mm

b

a

b

a

7. One as exponent: a1 = a

Ex a

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4.2 Negative Exponents and Scientific Notation

1. Negative exponent: n

n-

a

1a

2. Negative exponent fraction: n

m

m

-n

a

b

b

a

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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.

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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 Find the degree of the following terms: 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?

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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:

N = 2

nn2

Ex The height of a falling object is described the formula h = ho – 16t2 How far does a rock fall in 6 seconds?

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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:

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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: 4 2 1 X 3 5 2 Box Method Ex d

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4.6 Special Products Multiplying 2 binomials FOIL - First, Outer, Inner, Last

LL II OO FF (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 (A - B)2 A2 - B2 So (x - 1)2 x2 -1 Note: There is no formula that makes A2 + B2

38

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

Degree of Polynomial

Like terms must have exactly the same ___________________ Add/Subtract Polynomials Multiply Polynomials

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4.8 Division of Polynomials Cancellation (monomials): Dividing Polynomials by Monomials

Rule: x

d

x

c

x

b

x

a

x

d c b a

(Rule also applies for subtraction)

Dividing Polynomials by Binomials – use long division Recall arithmetic:

7 8 3 2 1

quotient

dividenddivisor

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.

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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: 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

41

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

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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 (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

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

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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 ca 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

44

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

x2 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

45

5.5 Factoring Sums and Difference of Cubes

x 1 2 3 4 5 6 10 1/2 0.1

x3 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

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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?

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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. How what is the maximum number of solutions of a quadratic equation? How many solutions could a cubic equation potentially have?

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5.8 Solving Applications of Quadratic Equations Numbers 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. 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) 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?

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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 5)1)(x(x

3x

undefined?

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)

Q

P

QK

PK where K 0

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6.2 Multiplication and Division of 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

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6.3 Adding and Subtracting; Least Common Denominator (LCD)

Rules: Q

RP

Q

R

Q

P and

Q

RP

Q

R

Q

P for Q 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 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. Compare the old denominator with the LCD to find “what’s missing” 2. Multiply top and bottom by the missing factor

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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.

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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

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6.6 Solving Rational Equations 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. 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

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6.7 Applications (these are in reverse order from the book) Proportions proportion - 2 ratios that equal each other e.g.

cross multiplying: If d

c

b

a then ad = bc (changes fractional equations to nonfractional)

Solving Proportion Equations Ex b 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 50

9

3

20 with

50

9

3

20

2) You can't cancel when terms are connected by addition Word Problem Chart

Quantity 1 Quantity 2

Case 1

Case 2

OR

Case 1 Case 2

Quantity 1

Quantity 2

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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)

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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.

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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)

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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)

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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 Interest Mixture

61

Distance/Rate/Time Wind/Current Let s = speed of plane in still air (the speed produced by the plane alone) w = wind speed s + w = speed with the wind s – w = speed against the wind

62

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”

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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

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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

a m/n = mnn m aa

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: nmnm aaa

Quotient rule: nm-

n

m

a a

a

Power rule: mnnm a)(a

Product to a power rule: (ab)m = ambm

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10.3 Multiplying Radical Expressions Multiplying: Product Rule (also applies for higher roots)

abba and baab 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)

66

10.4 Dividing Radical Expressions Dividing: Quotient Rule

b

a

b

a and

b

a

b

a for real numbers “a” and “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

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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:

baba

Radicals are similar to variable like terms – only like types can be added/subtracted Multiplying – distributive law Multiplying – FOIL

68

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

69

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"

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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

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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

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

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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:

x = 2a

4acbb 2 or x =

2a

4acbb 2

Putting the 2 together:

x1,2 = 2a

4acbb 2

Your book derives this formula. We won't!

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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:

P = tr)(1

A

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:

P = 2r)(1

A

r 1

A

; P = principal invested, A = amount paid out, r = interest rate

Solution: From our data, P = 11,000 and A = 6000. Plugging into the formula:

11,000 = 2r)(1

6000

r 1

6000

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

By the quadratic formula, r = 2(11)

1)4(11)(1616 2 = 0.06 or -1.5

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).

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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

g

lT 2

Motion Formulas

2

2

1attvs

atvv

o

of

Ex A cliff is 256 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?

75

11.5 Equations Reducible to Quadratics “Quadratic-like” Equations Quadratics that “pop out” of radical equations Quadratics from Rational Equations

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11.6 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

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