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P.1 Algebraic Expressions and 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?

- Students in biology, chemistry, physics.
- Students in chemistry.
- Students in calculus.
- 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

Example

- Simplify

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

Quotient Rule Explains Negative Exponent Rule

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?

When is your expression simplified?

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

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

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

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

Examples

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 youre taking the square root of the

same number can you add or subtract square roots

Simplify Expressions, then Add/Subtract (if

possible)

What is a conjugate?

- Pairs of expressions that involve the sum the

difference of two terms - The conjugate of ab 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

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!

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

Expressing roots as rational exponents

- Any root can be expressed as a rational exponent,

then rules of exponents apply

Expressions may involve exponents AND roots

- If possible, its often easier to take the root

first (the rational exponent), then raise the

value to the other exponent

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.

Special Products

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

Factor by Recognition

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

out

Factor Completely

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

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

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