Chapter P Pre-calculus notes

Prerequisites: Fundamental Concepts of Algebra Date:

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

Algebraic expression: a combination of variables and numbers using the operations of addition,

subtraction, multiplication, or division, as well as powers or roots

Examples:

Exponential notation: If n is a counting number (1,2,3, etc.), 𝑏𝑛 =

b is the _______________ and n is the _____________________

Evaluating an algebraic expression: find the value of the expression for a given value of the variable

Order of Operations

1. Start at the innermost set of parentheses and word outward.

2. Evaluate all exponential expressions.

3. Perform multiplication and division left to right.

4. Perform addition and subtraction left to right.

Ex. 1: Evaluate 8 + 6(𝑥 − 3)2 𝑓𝑜𝑟 𝑥 = 6.

When an equal sign is placed between two algebraic expressions, an _____________________ is

formed.

A formula is an ______________________ that uses variables to express a relationship between two or

more quantities.

Mathematical modeling: the process of finding formulas to describe real-world phenomena

Ex. 2: If the average cost of tuition and fees, T, for public four-year colleges, adjusted for inflation is

modeled by the formula 𝑇 = 17𝑥2 + 261𝑥 + 3257 where x is the number of years since the end of

the school year in 2000. Use the formula to project the average cost of tuition and fees at public U.S.

colleges for the school year ending 2010.

Set: a collection of objects whose contents can be clearly determined

The objects in a set are called the ________________________ of the set.

Ways to represent a set:

Roster method (listing all the elements, separated by commas.

Set-builder notation:

Intersection of sets A and B: the set of elements common to both set A and set B

Notation:

Venn diagram:

Ex. 3: Find the intersection: {3,4,5,6,7} ∩ {6,8,10,12}

Union of sets A and B: the set of elements that are members of set A or of set B or of both sets.

Notation:

Venn diagram:

Ex. 4: Find the union: {3,4,5,6,7} ∪ {3,7,8,9}

A set with no elements is called _______________________ or ________________________.

Hmk: Math XL: Log in and that complete “Day 1 assignment” to practice using the features of the program. Also use the code to view the e-book and complete the following assignment:

Pgs. 15-16: #3-48 (mult of 3)—check odd answers—make sure to write out the problems and show all

of your work.

Chapter P Pre-calculus notes

Prerequisites: Fundamental Concepts of Algebra Date:

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers (cont.)

Subsets of the Real Numbers

Name Symbol Description Examples

Natural Numbers

Whole Numbers

Integers

Rational Numbers

Irrational Numbers

Real numbers: the set of numbers that are either rational or irrational.

Ex. 5: Consider the following set of numbers: {−9, −1.3,0,0.3,𝜋

2, √9, √10}

List the members in the set that are:

Natural numbers

Whole numbers

Integers

Rational numbers

Irrational members

Real numbers

Ordering the real numbers

Symbols : ≤ and ≥

Absolute Value: |a|

Informal def:

Formal def: |𝑥| = {

Ex. 6: Rewrite without absolute value bars:

a) |1 − √2| b) |𝜋 − 3| c) |𝑥|

𝑥 if x>0.

Distance between Points on a Real Number Line: a and b are any two pts on a real number line, then

the distance between a and b is given by:

Ex. 7: Find the distance between -4 and 5. _________________ = _______________

Properties of Real Numbers and Algebraic Expressions

Name Meaning Example

Commutative Property of Addition/Multiplication

Associative Property of Addition/Multiplication

Distributive Property

Identity Property of Addition

Identity Property of Multiplication

Inverse Property of Addition

Inverse Property of Multiplication

Definition of Subtraction: a – b =

-b is called the _____________________________ of b.

Definition of Division: 𝑎 ÷ 𝑏 =

1

𝑏 is called the _______________________________ of b.

Simplifying Algebraic Expressions: Combine like terms (add their coefficients)

Ex. 8: Simplify: 6(2𝑥2 + 4𝑥) + 10(4𝑥2 + 3𝑥)

Properties of Negatives: Let a and b represent real numbers, variables, or algebraic expressions.

Property Example

(-1)a =

-(-a) =

(-a)b =

a(-b) =

-(a+b) =

-(a-b) =

If a negative sign or a subtraction symbol appears outside parentheses, drop the parentheses and

change the sign of every term within the parentheses. (Distribute the negative.)

Ex. 9: Simplify: 6 + 4[7 − (𝑥 − 2)]

Ex. 10: Name the property illustrated by each statement.

a) 11 ∙ (7 + 4) = 11 ∙ 7 + 11 ∙ 4

b) 7∙(11 ∙ 8) = (11 ∙ 8) ∙7

c) 1

𝑥+3(𝑥 + 3) = 1, 𝑥 ≠ −3

Pgs. 16-19: # 51-159 (mult of 3, skip 144, 147), 160-162 (all)—check odd answers

P.3 Radicals and Radical Expressions Pre-calculus notes Date:

Def: If 𝑏2 = 𝑎, 𝑡ℎ𝑒𝑛 𝑏 is a ______________ _______________ of a.

Symbol: _______ represents the positive or ________________________________________ of a

number.

The symbol ________ is called a _______________ _______________ and the number under the sign

is called the ____________________. Together they form a ________________ ________________.

Ex. 1: Evaluate:

a) √81 = b) −√9 = c) √1

25=

d) √36 + 64 = e) √36 + √64 =

Note: The square root of a negative number is NOT a real number.

Def: For any real number a, √𝑎2 =

Product Rule for Square Roots: If a and b represent non-negative real numbers,

then _______________________ and __________________________.

A square root is simplified when:

Ex. 2: Simplify.

a) √75 b) √5𝑥 ∙ √10𝑥

Quotient Rule for Square Roots

If a and b represent non-negative real numbers and 𝑏 ≠ 0,

then _________________________ and _________________________.

Ex. 3: Simplify.

a) √25

16= b)

√150𝑥3

√2𝑥

Adding and Subtracting Square Roots:

Two or more radicals can be added/subtracted only if they are ______________________________.

Ex. 4: Add or subtract as indicated.

a) 8√13 + 9√13 = b) √17𝑥 − 20√17𝑥 =

Ex. 5: Add or subtract as indicated. It may be necessary to simplify first.

a) 5√27 + √12 b) 6√18𝑥 − 4√8𝑥

A radical expression is not simplified if there is a radical in the denominator. The process of rewriting

the expression to eliminate any radicals in the denominator is called

_______________________ the ___________________________.

Ex. 6: Simplify.

a) 5

√3 b)

6

√12

Def: √𝑎 + √𝑏 and √𝑎 − √𝑏 are called _________________________________.

Multiply them together and the result is:

Ex. 7: Rationalize the denominator—Use the conjugate!

8

4 + √5

Other kinds of roots:

The principal nth root of a real number a, symbolized by ______________ is defined by the

following:

n is called the _________________

If n is even, then a and b are ________________________.

If n is odd, then a and b ___________________________________.

Cube Roots Fourth Roots Fifth Roots

The product and quotient rules apply to all higher roots as well.

Ex. 8: Simplify.

a) √403

b) √85

∙ √85

c) √125

27

3

Rational Exponents:

Def: If √𝑎𝑛

represents a real number where n_________ is an __________________, then

√𝑎𝑛

=

Also, 𝑎−1

𝑛 =

Ex. 10: Simplify.

a) 251

2 = b) 81

3 = c) −811

4 =

d) (−8)1

3 = e) 27−1

3 =

Def: If √𝑎𝑚𝑛 represents a real number and

𝑚

𝑛 is a _________________________________, where

n ___________, then 𝑎𝑚

𝑛 =

Also, 𝑎−𝑚

𝑛 =

Ex. 11: Simplify.

a) 274

3 b) 43

2 c) 32−2

5

Ex. 12: Simplify using properties of exponents.

a) (2𝑥4

3) (5𝑥8

3) b) 20𝑥4

5𝑥32

Rational exponents are sometimes useful for simplifying radicals by reducing the index.

Ex. 13: Simplify: √𝑥36.

Homework: Pgs. 46-48: #3-108 (mult of 3), 123, 125, 133, 137

P.4 Polynomials Pre-calculus notes Date:

A polynomial is a single term or the sum of two or more terms containing variables with whole number

exponents.

The standard form of a polynomial is found by writing the terms in __________________________

powers of the variable.

If 𝑎 ≠ 0, the degree of 𝑎𝑥𝑛 is _______. The degree of a nonzero constant is ________.

A polynomial with one term is called a _____________________.

A polynomial with two terms is called a ____________________.

A polynomial with three terms is called a ______________________.

The degree of a polynomial is _________________________________________________.

Definition of a Polynomial in x

A polynomial in x is an algebraic expression of the form

Where ______________________________________ are real numbers and 𝑎𝑛 ≠ 0 and n is a non-

negative integer. The polynomial is of degree ______ , 𝑎𝑛 is the _______________________

_________________________, and 𝑎0 is the _________________________ ________________.

Adding and Subtracting Polynomials

Ex. 1 Perform the indicated operations and simplify.

a) (−17𝑥3 + 4𝑥2 − 11𝑥 − 5) + (16𝑥3 − 3𝑥2 + 3𝑥 − 15)

b) (13𝑥3 − 9𝑥2 − 7𝑥 + 1) − (−7𝑥3 + 2𝑥2 − 5𝑥 + 9)

Multiplying Polynomials: Use properties of exponents.

To multiply polynomials when neither is a monomial, multiply each term of one polynomial by each

term of the other polynomial. Then combine like terms.

Ex. 2: Multiply: (5𝑥 − 2)(3𝑥2 − 5𝑥 + 4)

To multiply two binomials, FOIL (First-Outer-Inner-Last)

Ex. 3: Multiply: (7𝑥 − 5)(4𝑥 − 3)

Special Products

1) Product of Sum and Difference of Two Terms

(𝑎 + 𝑏)(𝑎 − 𝑏) =

2) The Square of a Binomial Sum

(𝑎 + 𝑏)2 =

3) The Square of a Binomial Difference

(𝑎 − 𝑏)2 =

4) The Cube of a Binomial Sum

(𝑎 + 𝑏)3 =

5) The Cube of Binomial Difference

(𝑎 − 𝑏)3 =

Ex. 4: Multiply.

a) (7𝑥 + 8)(7𝑥 − 8) =

b) (2𝑦3 − 5)(2𝑦3 + 5) =

Ex. 5: Multiply.

a) (𝑥 + 10)2 =

b) (5𝑥 + 4)2 =

Ex. 6: Multiply.

a) (𝑥 − 9)2 =

b) (7𝑥 − 3)2 =

Polynomials in two variables x and y.

The degree of 𝑎𝑥𝑛𝑦𝑚 is _______________________.

Ex. 7: Subtract: (𝑥3 − 4𝑥2𝑦 + 5𝑥𝑦2 − 𝑦3) − (𝑥3 − 6𝑥2𝑦 + 𝑦3).

Ex. 8: Multiply.

a) (7𝑥 − 6𝑦)(3𝑥 − 𝑦) b) (2𝑥 + 4𝑦)2

Ex. 9: Perform the indicated operations.

a) (3𝑥 + 4𝑦)2 − (3𝑥 − 4𝑦)2 b) (5𝑥−3)6

(5𝑥−3)4

Homework: Pgs. 58-59: #3-81 (mult of 3), 91, 94, 107, 108

Quiz over P.1-P.4 this Thursday 9/1

P.5 Factoring Polynomials Pre-calculus notes Date:

Factoring a polynomial containing the sum of monomials means finding an equivalent expression that is

a product.

We will be factoring over the set of integers. All coefficients will be integers.

Polynomials that cannot be factored using integer coefficients are called _____________________ over

the integers or _________________.

The first step in any factoring problem is to look for the _____________ _____________ __________.

Ex. 1: Factor.

a) 10𝑥3 − 4𝑥2 b) 2𝑥(𝑥 − 7) + 3(𝑥 − 7)

When factoring a polynomial with four terms it may be possible to factor by grouping.

Ex. 2: Factor: 𝑥3 + 5𝑥2 − 2𝑥 − 10.

Factoring trinomials of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐

Find two First terms whose product is 𝑎𝑥2.

Find two Last terms whose product is c.

By trial and error try all arrangements of the factors until the sum of the Outside product

and the Inside product is 𝑏𝑥.

If no such combination exists, the polynomial is ________________.

Ex. 3: Factor: 𝑥2 + 13𝑥 + 40.

Ex. 4: Factor: 𝑥2 − 5𝑥 − 14.

Ex. 5: Factor: 6𝑥2 + 19𝑥 − 7.

Ex. 6: Factor: 3𝑥2 − 13𝑥𝑦 + 4𝑦2.

Factoring the Difference of Two Squares: 𝑎2 − 𝑏2 =

Ex. 7: Factor:

a) 𝑥2 − 81 b) 36𝑥2 − 25

The goal of factoring is to factor COMPLETELY.

Ex. 8: Factor (completely): 𝑥4 − 81.

Factoring Perfect Square Trinomials

𝑎2 + 2𝑎𝑏 + 𝑏2 =

𝑎2 − 2𝑎𝑏 + 𝑏2 =

In a perfect square trinomial:

The first and last terms are squares of monomials or integers.

The middle term is twice the product of the expressions being squared in the first and last

terms.

Ex. 9: Factor.

a) 𝑥2 + 6𝑥 + 9 b) 25𝑥2 − 60𝑥 + 36

Factoring Sums and Differences of Cubes

𝑎3 + 𝑏3 =

𝑎3 − 𝑏3 =

Ex. 10: Factor.

a) 𝑥3 + 1 b) 125𝑥2 − 8

Refer to the strategy for factoring polynomials on Pg. 69 in the textbook.

Ex. 11: Factor: 3𝑥3 − 30𝑥2 + 75𝑥 Start with the GCF !

Factoring by grouping can include grouping a trinomial.

Ex. 12: Factor: 𝑥2 − 36𝑎2 + 20𝑥 + 100.

Factoring with fractional and negative exponents.

Recall: Expressions containing fractional exponents are NOT polynomials. They can be simplified using

factoring techniques.

Ex. 13: Factor and simplify: 𝑥(𝑥 − 1)−1

2 + (𝑥 − 1)1

2.

Quiz Thursday over sections P.1-P.4.

Homework: Math XL assignment: P.1-P.4 review (due by class time)

Due Friday (sect. P-5): Pgs. 71-73: #3-99 (mult of 3), 134-137 (all)

*You may work on the assignment due Friday after you finish your quiz.—I will be checking for all of

the supplies (3-ring binder, graph paper, calculator) during the quiz on Friday.

P.6 Rational Expressions Pre-calculus notes Date:

A rational expression is the __________________ of two polynomials.

The set of real numbers for which an algebraic expression is defined is the _______________ of the

expression.

Because division by zero is undefined, all values that make the denominator zero must be excluded.

Ex. 1: Find all the numbers that must be excluded from the domain of each rational expression:

a) 7

𝑥 + 5 b)

𝑥

𝑥2 − 36

A rational expression is simplified if its numerator and denominator have no common factors other

than 1 and -1. To simplify a rational expression:

Factor the numerator and the denominator completely.

Divide both the numerator and denominator by any common factors.

Ex. 2: Simplify:

a) 𝑥3+3𝑥2

𝑥+3 b)

𝑥2−1

𝑥2+2𝑥+1

To multiply rational expressions:

Factor all numerators and denominators completely.

Divide numerators and denominators by common factors.

Multiply the remaining factors in the numerators and multiply the remaining factors in the

denominators.

Ex. 3: Multiply: 𝑥+3

𝑥2−4∙

𝑥2−𝑥−6

𝑥2+6𝑥+9.

To divide two rational expressions, the problem must be rewritten as a multiplication problem by

using the reciprocal

Ex. 4: Divide: 𝑥2−2𝑥+1

𝑥3+𝑥÷

𝑥2+𝑥−2

3𝑥2+3

To add and subtract rational expressions they must have common denominators. Be careful to use

the distributive property correctly.

Ex. 5: Subtract: 𝑥

𝑥+1−

3𝑥+2

𝑥+1

Ex. 6: Add: 3

𝑥+1+

5

𝑥−1

To find the least common denominator, or LCD:

Factor each denominator completely.

List all the factors of the first denominator.

Add to the list in set 2 any factors of the second denominator that do not appear in the list.

Find the product of the factors in Step 3.

Ex. 7 Find the LCD of the following rational expressions:

3

𝑥2−6𝑥+9 𝑎𝑛𝑑

7

𝑥2−9

Ex. 8: Subtract: 𝑥

𝑥2−10𝑥+25−

𝑥−4

2𝑥−10.

_____________________ __________________ have numerators and denominators containing one

or more rational expressions. (A fraction within a fraction.) There are two methods for simplifying.

Method #1: Combine the numerator into a single expression and combine the denominator into a single

expression. Then perform the division by rewriting as a multiplication problem.

Ex. 9: Simplify using Method #1. 1

𝑥−

3

21

𝑥+

3

4

.

Method #2: Find the least common denominator of all the rational expressions in the numerator and

denominator. Then multiply each term in the numerator and denominator by this LCD. (Preferred

method.)

Ex. 10: Simplify using Method #2. 1

𝑥+7−

1

𝑥

7.

Homework due Tuesday: Pgs. 82-83: #3-69 (mult of 3)

Homework due Wednesday: Pg. 88: #1-32

Homework due Thursday by class time: Math XL: Ch. P Review

Ch. P Test on Thursday 9/8

Homework due Friday by class time: Math XL: Ch. 1 Preview