Chapter P

Prerequisites: Fundamental Concepts of Algebra

P.1 Algebraic Expressions & Real Numbers

Objectives• Evaluate algebraic expressions• Use mathematical models• Find the intersection of 2 sets• Find the union of 2 sets• Recognize subsets of the real numbers• Use Inequality symbols• Evaluate absolute value• Use absolute value to express distance• Identify properties of the real numbers• Simplify algebraic expressions

Intersection of Sets

• What they have in common

• A = {all tall children}

• B = {all girls}

• A intersect B = {all tall girls}

• All children that are girls AND are tall

Union of Sets

• Combination of everything in both sets• A = {all tall children}• B = {all girls}• A union B = {all girls OR tall children} = {all girls

and all tall boys}

A group of biology majors are taking Biology I & Chem. I. A group of chemistry majors are taking Calculus, Chem. I and

Physics I. The Physics majors enrolled in Calculus, Physics I, and Chem I. What is

the intersection of the 3 groups?

1. Students in biology, chemistry, & physics.

2. Students in chemistry.

3. Students in calculus.

4. Students in physics.

Absolute Value

• │x│ represents the distance between x and zero• Distance is always a positive quantity, therefore

going left or right x units results in a distance of x units

• │x - 2│ represents the distance between x and 2• Distance is again always positive. (i.e. the

distance between 2 and 3 is 1 and the distance between 2 and 1 is 1) │3 - 2│ = │1 - 2│ = 1

Real numbers are a field• Commutative (addition & multiplication)• Associative (addition & multiplication)• Identity (additive = 0 & multiplicative = 1)• Inverse (additive = -x & multiplicative = 1/x)• Distributive (multiplication over addition)

• ALL these properties are useful when manipulating algebraic expressions & equations

P.2

• Exponents and Scientific Notation

Objectives

• Use the product and quotient rules• Use the zero-exponential rule• Use the negative-exponent rule• Use the power rule• Find the power of a product• Find the power of a quotient• Simplify exponential expressions• Use scientific notation

Product & Quotient Rules

332

5

743 )()(

aaa

aaa

aa

aaaaa

a

a

xxxxxxxxxx

Example

• Simplify:

abcbacb

cab

ab

cb

cbba

cb

cab

2

3

16

24

16

24

2

3

28

38

16

24

)55()23(52

53

52

52

52

53

Quotient Rule explainszero-exponent rule

• Any real number divided by itself (except 0) equals 1

• If x is any nonzero number & y is an exponent:

1

1

0

0)(

x

x

x

xxx

x

y

y

yyy

y

Quotient Rule Explains Negative Exponent Rule

22

223

3

5

3

2)53(5

3

1

1

xx

xxx

x

x

x

xxx

x

Working with Negative Exponents

• In general, expressions are not considered simplified when negative exponents are present.

• A negative exponent in the denominator becomes positive when moved to the numerator

• A negative exponent in the numerator becomes positive when moved to the denominator

Raising an Exponent to an Exponent (Power Rule)

• Exponents are multiplied• WHY?

nmnm xx

bababababa

)(

))()(()( 96323232332

When is your expression simplified?

• No negative exponents are present• Each base appears only once• No parentheses remain• Example:

8

81

8

81

)2(

)3( 185

63

128

32

432 yx

yx

yx

xy

yx

Scientific Notation

• What is it? A number greater than or equal to 1 & less than 10 (either pos. or neg.) multiplied by 10 raised to an exponent

• Example:

7

5

1013.5

101.2

Why Use Scientific Notation?

• It allows us to express very large numbers or very small numbers in a more concise manner.

• Diminishes the error in writing very large or small numbers by eliminating the need to have all zeros written. (easy to have one too many or too few zeros)

Converting into Scientific Notation

46

2

5

1091.610

1091.6

000,000,1

691000691.

1045.3000,10045.3000,345

Rules of Thumb

• Count decimal places you move to place the decimal to the right of one non-zero digit

• Large numbers are represented by multiplying by ten raised to a positive exponent

• Small numbers are represented by multiplying by ten raised to a negative exponent

P.3

• Radicals & Rational Exponents

Objectives

• Evaluate square roots• Simplify (nth root of nth power)• Use product & quotient rules to simplify square

roots• Add & subtract square roots• Rationalize denominators• Evaluate & perform operations with higher roots• Understand & use rational exponents

Principal Square Root• It is true that 4 squared and (-4) squared both

equal 16, BUT the principal square root of 16 is 4 NOT -4

• By convention, the radical symbol represents the positive (or PRINCIPAL) square roots of a number, thus for real numbers, x, greater than or equal to 0:

0x

Examples

2866436

3

1

9

1

1051025250

Multiplying & Dividing with Radicals (Roots)

• A product or quotient under a radical can be written as the product or quotient of separate radicals

• Products or quotients involving square roots can be expressed as a single square root involving products or quotients under the radical

Adding & Subtracting Square Roots

• ONLY when you’re taking the square root of the same number can you add or subtract square roots

752

777572

Simplify Expressions, then Add/Subtract (if possible)

611615646)3(56)2(2

695642545242

What is a conjugate?

• Pairs of expressions that involve the sum & the difference of two terms

• The conjugate of a+b is a-b

• Why are we interested in conjugates?

• When working with terms that involve square roots, the radicals are eliminated when multiplying conjugates

Multiplying conjugates which involve square roots

yxyx

yyxyxyxx

yxyx

22

)()(

Expressions with radicals in the denominator are NOT simplified

• Eliminate the radical from the denominator by multiplying by the numerator and the denominator by the conjugate of the denominator

• Sometimes the result may not LOOK simpler!

6

6233618

39

6233618

)33(

)33(

33

26

33

26

Other Roots

• The nth root of a number means “what number could you raise to the nth power to get your original number?”

• You can take an odd root of a negative number or a positive number.

• You can only take an even root of a positive number.

Rules for other roots

• Add and subtract only same roots of same number (i.e. you can add cube roots of 3 but NOT cube roots of 3 and cube roots of 4)

• Multiply & divide same roots following same rules as square roots

Subtract, if possible, & simplify:

3

3

3

3

28)4

24)3

22)2

142)1

33 24166

Expressing roots as rational exponents

• Any root can be expressed as a rational exponent, then rules of exponents apply

nn xx1

Expressions may involve exponents AND roots

• If possible, it’s often easier to take the root first (the rational exponent), then raise the value to the other exponent

32)2()16(16 5544

5

P.4

• Polynomials

Objectives

• Understand the vocabulary of polynomials

• Add & Subtract polynomials

• Multiply polynomials

• Use FOIL in polynomial multiplication

• Use special products in polynomial multiplication

• Perform operations with polynomials in several variables

• A polynomial in x is many terms added or subtracted with each term involving a constant and x raised to a power.

• Only same powers of x can be added/subtracted

• When multiplying polynomials, the distributive property holds. (i.e. every term in one polynomial must be multiplied by every term in the other polynomial.

)723)(53( 3 xxx

35316159

3510152169

)723(5)723(3

234

324

33

xxxx

xxxxx

xxxxx

Special Products

3223

3223

22

22

22

33))()((

33))()((

2))((

2))((

))((

babbaabababa

babbaabababa

babababa

babababa

bababa

P.5

• Factoring Polynomials

Objectives

• Factor out the greatest common factor• Factor by grouping• Factor trinomials• Factor difference of squares• Factor perfect square trinomials• Factor sum & difference of cubes• Use a general strategy for factoring• Factor expressions containing fractional &

negative exponents

Factoring strategies

• FIRST: Look for greatest common factor

• Group terms (if 4 or more) to find common terms between groups

• If only 3 terms, rewrite into 4 terms by multiplying leading coefficient by the constant term (a times c), then rewrite bx as the sum of 2 terms whose product of their coefficients is ac (then group as in previous item)

EXAMPLE

)43)(52(

)52(4)52(3

208156

,23815

815120

120206

20236

2

2

xx

xxx

xxx

substitutexxx

xx

Factor by Recognition

• Difference of Squares• Difference or Sum of Cubes• Opposite signs cause all middle terms to cancel

out

))((

))((

))((

2233

2233

22

yxyxyxyx

yxyxyxyx

yxyxyx

Factor Completely

)42)(2(4)4

)42)(2(4)3

)8(4)2

)8)(44)(1

22

22

33

22

yxyxyx

yxyxyx

yx

yxyx

33 324 yx

P.6

• Rational Expressions

Objectives

• Specify domain of a rational expression

• Simplify rational expressions

• Multiply rational expressions

• Divide rational expressions

• Add & subtract rational expressions

• Simplify complex rational expressions

• Simplify fractional expressions that occur in calculus

• Rationalize numerators

Domain restrictions

• No values can be substituted in for x that would create a zero denominator or a negative value under a positive root

Simplify rational expressions

• Factor numerator and denominator to cancel common terms

• Do NOT forget that the terms cancelled still were in the original expression, therefore must be considered when stating the domain

Adding & Subtracting Rational Expressions

• Expressions MUST have a common denominator to be added/subtracted

• Remember when creating a common denominator, both the numerator & denominator must be multiplied by the same term, otherwise the resulting expression will NOT be equivalent to the original

Simplify

xx

xx

xx

xxx

x

xx

x

x

xx

x

x

xx

xx

x

x

x

x

x

xx

x

x

x

x

x

xx

x

2

13

)2(

21

2

2

)2(

1

2

2

)2(

1

21

2)(

2

1

)(

21

2

2

1

2

23232

2222

Rational expressions that occur in calculus

• To simplify this expression, you may have to rationalize the numerator (if f(x) involves a root)

h

xfhxf )()(

11

1

)11()11(

)1()1(11

1111:

11)()(

1)(,1)(

xhxxhxh

h

xhxh

xhxxhx

xhx

h

xhxerationaliz

h

xhx

h

xfhxf

hxhxfxxf

P.7 Equations

• Objectives

• Solve linear equations in one variable

• Solve linear equations containing fractions

• Solve rational equations with variables in the denominators

• Solve a formula for a variable

• Solve equations involving absolute value

Objectives continued

Solve quadratic equations by:a) Factoringb) Using the square root propertyc) Completing the squared) Using the quadratic formula

(WHEN TO USE WHICH METHOD?)Use discriminant to determine # & type of

solutionsSolve application problems involving

quadraticsSolve radical equations

What is a linear equation in one variable and what is “solving it”?

Only one variable (x or y, generally) is in the equation and it is NOT squared or raised to a power other than 1.

To “solve” the equation means to find the value (or values) that would make the equation true.

How do we solve an equation?

• Eliminate parentheses (distribute!)• Collect like terms (additive identity)• Isolate the variable (multiplicative identity)• Remember: it’s an EQUATION to start

with, meaning the left equals the right. It will no longer be equal, if something is done to one side and not the other!

• CHECK your solution in the original equation: does it make it true?

EXAMPLE• 4(2x-3) = 2(x+3)• 1)Distribute to eliminate parentheses

8x-12 = 2x + 6 2)Collect x’s on one side & constants on

the other (use additive identity) 8x(-2x) – 12(+ 12) = 2x(-2x) + 6(+ 12) 6x = 18 3)Isolate the x (use multiplicative identity) 4) Check your solution in the original

4(2(3)-3) = 2(3+3) 4(3) = 2(6) YES!!

36

18

6

6

x

x

Rational Equations

• Equations that involve fractions!• The variable (x) could be in the numerator of the

denominator. IF the x is found in the denominator, we must consider values x canNOT take on. (i.e. zero denominator)

• EVEN after you’ve simplified an equation to eliminate the fractions, you haven’t eliminated the original restriction that may have been present.

• With fractions, EITHER eliminate the fraction OR get a common denominator (if denominators are EQUAL, so are numerators)

Solve by getting a common denominator

!!

6,318

884826

8426

)2(8

84

)2(8

224

)4)(2(2

)4)(2(1

)2(8

)2(1

)8)(2(

)8(3

??,...2,2

1

8

1

2

3

CHECK

xx

xxxx

xx

x

x

x

x

x

x

x

x

x

WHYxx

Types of Equations

• Conditional: True under certain conditions (could be one or several solutions)

• Inconsistent: Inconsistencies between the 2 sides (never true – NO solutions)

• Identity: One side of the equation is identical to the other (doesn’t matter what x is, infinitely many solutions)

Example

• Solve 3x – 6 = 3(x – 2)Notice, after distributing on the

right, 3(x – 2) = 3x – 6The left side is identical to the

right. No matter what values you plug in for x, it will always be true.

The solution set is: {all reals}

THIS IS AN IDENTITY.

Example

• Solve: 4x – 8 = 4(x – 5)• Distribute on right = 4x – 20• Think: Can 4 times a number minus 8

possibly equal 4 times the same number minus 20??? NO!!

• If you continue to solve, you get:• 0x = -12 (Can 0 times a number ever equal -12?

NO!• INCONSISTENCIES!! Solution: { }

Quadratic equations

Various methods to solve different quadratics

What is a quadratic equation?

}{Re,,

02

alscba

cbxax

Zero-Product Rule

• If the product of two or more numbers is zero, at least one of the numbers must equal zero!

• If AB=0, then A=0 and/or B=0– One or both of the terms must equal zero

Why is this important?

It allows us an easy way to solve an equation, but FIRST make certain the expression is a product that equals zero.

• A product involves FACTORS

• (2x-3)(x+2)=0

• 2x – 3 is a factor of the expression, as is 2+x

• Set each factor = 0

• 2x – 3 = 0, thus x = 3/2

• x + 2 = 0, thus x = -2

• SO, if EITHER x = 3/2 or x = -2, the original expression = 0

• SO, solve by FACTORING if equation, once equal to 0, is FACTORABLE

Often, you must get expression into factored form FIRST:

2

1,3

4

012

043

0)12)(43(:

04116

4116:2

2

x

x

x

xxFactor

xx

xxSolve

Solving with square root property

• When would you use this approach?– When one side of the equation is a perfect squareEXAMPLE:

ii

x

ix

WHYix

Whyx

x

2

5

2

3

2

53

532

??)(523

??)(5)23(

5)23(

2

2

Solve by Completing the Square

• When can you use this method? ALWAYS– However, if the expression is factorable or is

already a perfect square, those methods may be more desirable

HOW does it work?

If you don’t have a perfect square, you create one by adding a “well-chosen” zero (adding the same term to both sides)

Decide what to add by determining what additional term would create a perfect square

EXAMPLE

ixix

x

x

xx

xxxNote

xx

xx

xxSolve

2

51,

2

51

2

5)1(

2

5)1(

2

51

2

712

)12)1(:(

2

72

0)2

72(2

0742:

2

2

2

22

2

2

2

Completing the square generalized to any quadratic equation results in

the quadratic formula.• When can you use it? ALWAYS. (However, it still

may be easier to factor & use zero-factor property or take the square root if it’s already a perfect square.)

a

acbbx

cbxax

2

4

0

2

2

Solve:

1. x = -1, 3

2. x = 1, -3

3. x = 2,3

4. x = 2

642 2 xx

What is the discriminant and why is it useful to us?

• The discriminant is the part of the quadratic equation that is under the radical.

• Based on what is under that radical, we can determine if our solution will be an integer (is what’s under there a perfect square?), an irrational (is what’s under there a positive number that is NOT a perfect square), or complex (is what’s under there a negative number?)

What if your equation involves a variable under a radical?

• In order to eliminate an nth root, you must raise both sides of the equation to the nth power.

• Be CERTAIN that you isolate the radical (have it on one side of the equation by itself) before you raise both sides to the nth power.

What if the variable is found under a radical twice in an equation?

• Isolate one radical and raise both sides to the nth power.

• Then, isolate the other radical (it will not have disappeared from the other side), and raise both sides to the nth power again.

What is x is raised to an exponent that is NOT an integer?

• If the variable (or expression involving a variable) is raised to the (m/n) exponent, you must isolate that expression and then raise BOTH sides to the (n/m) power.

• WHY?? When you raise one exponent to another, you multiply the 2 exponents.

baba

ba

m

n

n

m

n

m

))((

)(

What if the equation involves an expression inside absolute value

brackets?• Recall what absolute value means: What is

within those brackets could be positive or negative and still have the same overall value.

3

8,83

1573

15)73()3

22,223

1573)

1573

xx

x

xII

xx

xI

x

Solve:

1. No solution.

2. {7/3}

3. {10/3, 4/3}

4. {-7/3, 7/3}

1273 x

P.8

• Modeling with Equations

Objectives

• Use equations to solve problems

Solving Word Problems

• 1) Carefully read the problem• 2) Determine what do you know and

what do you want to know• 3) Identify variables• 4) Develop equation relating what you

know & what you want to know• 5) Solve the equation & check (correct?)• 6) Make certain you answered the

question you were being asked!

EXAMPLE

• You need to drive from Chicago to your cousin’s house in Omaha (a distance of 550 miles) at an average 65 mph on the Interstate highway. What time should you leave if you have to be at your cousin’s at 3:30 pm?

• What do you want to know?– How long will it take you to drive? (x = time)– What time must you leave?

• What do you know?– Total distance you’ll travel (550 miles)– Speed (65 miles per hour)

• What is the relationship between known & unknown?– Distance = Rate x Time– 550 miles = 65 mph x (X) (cont. on next slide)

Did you answer the question?NO – WHEN should you leave?In order to arrive at 3:30pm, you leave 8.5 hrs earlier, which would

be at 8:00 am.

hr

hrmimi

x 5.865

550

EXAMPLE

• You have been asked to make an aluminum can (cylindrical shape) to hold 300 ml of your product. The can is to be 10 cm high. How much aluminum (in square cm) do you need?

• What do you know?– Can holds 300 ml (the volume!)– Height = 10 cm

What do you want to know?-How much material you will need (surface area).

What relates the known & unknown?

For cylinders:

rhrSA

hrV

22 2

2

(example continued)

cmr

cmcm

cmr

cmrcmmlV

1.3

55.910

300

)10(300300

23

2

23

Now find surface area!(answer the question!)

(remember, a cylinder is just 2 circles and a rectangle)

2

2

2

255

)10)(1.3(2)1.3(2

22

cmSA

cmcmcmSA

rhrSA

Ava purchased a new ski jacket, on sale for $66.50. The coat had been advertised as 30% off! What was

the original cost?

1. $95

2. $86.50

3. $90

4. $96.50

P.9

• Linear Inequalities and Absolute Value Inequalities

Objectives

• Use interval notation.

• Find intersections & unions of intervals.

• Solve linear inequalities.

• Recognize inequalities with no solution or all numbers as solutions.

• Solve compound inequalities.

• Solve absolute value inequalities.

Linear inequalities

• For equalities, you are finding specific values that will make your expression EQUAL something. For inequalities, you are looking for values that will make your expression LESS THAN (or equal to), or MORE THAN (or equal to) something.

• In general, your solution set will involve an interval of values that will make the equation true, not just specific points.

What if you have more than one inequality?

• If two inequalities are joined by the word “AND”, you are looking for values that will make BOTH true at the same time. (the INTERSECTION of the 2 sets)

• If two inequalities are joined by the word “OR”, you are looking for values that will make one inequality OR the other true (not necessarily both), therefore it is the UNION of the 2 sets.

What IS an absolute value inequality?

• Recall that absolute value refers to the expression inside the brackets being either positive or negative, therefore the absolute value inequality involves 2 separate inequalities

• IF absolute value expression is LESS THAN a value, you’re looking for values that are WITHIN that distance (intersection of the 2 inequalities)

• IF absolute value expression is MORE THAN a value, you’re looking for values that are getting further away in both directions (union of the 2 inequalities)

If the absolute value is greater than a number, you’re considering getting further away in both directions, therefore an OR. (get further away left OR right)

• See next slide for example:

2

7

2

52

5

382

3)42(2

7

342

342

xx

x

x

xOR

x

x

x

• If, however, the absolute value was LESS than a number (think of this as a distance problem), you’re getting closer to your value and staying WITHIN a certain range. Therefore, this is an intersection problem (AND)

• Same problem as before, but solved as a LESS than inequality. (next slide)

2

7,2

5,2

7

2

52

5

382

3)42(2

7

342

342

xx

x

x

xAND

x

x

x

Don’t leave common sense at the door!

• Remember to use logic!

• Can an absolute value ever be less than or equal to a negative value?? NO! (therefore if such an inequality were presented, the solution would be the empty set)

• Can an absolute value ever be more than or equal to a negative value?? YES! ALWAYS! (therefore if such an inequality were given, the solution would be all reals)