Review of College Algebra: Sets, Real Numbers, and Polynomials

R.1 Sets & Notation

Set Theory and Notation

Set theory is a foundational concept in mathematics, involving the study of collections of objects, called sets. Understanding sets and their operations is essential for higher-level mathematics, including trigonometry.

• Set: A collection of distinct objects, denoted by curly braces { }.

• Element: An object or member of a set, denoted by the symbol ∈ (e.g., 4 ∈ A means 4 is in set A).

• Finite Set: Contains a limited number of elements.

• Infinite Set: Contains an unending list of elements.

• Set Builder Notation: Describes a set by a property its members satisfy, e.g., { x | x is a natural number less than 5 } = {1, 2, 3, 4}.

• Natural Numbers (\mathbb{N}): {1, 2, 3, 4, ...}

• Universal Set (U): The set of all elements under consideration in a particular context.

• Null or Empty Set (\emptyset): A set with no elements.

• Subset (\subseteq): A set whose elements are all contained within another set.

Example: If A = {2, 5, 9} and B = {2, 3, 4, 5, 6, 9, 10}, then A \subseteq B but B \nsubseteq A.

Venn Diagrams

Venn diagrams visually represent relationships between sets, such as intersections, unions, and complements.

Operations on Sets

• Complement (A'): Elements in the universal set U but not in set A.

• Intersection (A \cap B): Elements common to both sets A and B. Set Builder Notation: { x | x ∈ A and x ∈ B } Example: {9, 15, 25, 36} \cap {15, 20, 25, 30, 35} = {15, 25}

• Disjoint Sets: Sets with no elements in common (A \cap B = \emptyset).

• Union (A \cup B): All elements in either set A or set B (or both). Set Builder Notation: { x | x ∈ A or x ∈ B } Example: {9, 15, 25, 36} \cup {15, 20, 25, 30, 35} = {9, 15, 20, 25, 30, 35, 36}

R.2 Real Numbers and Their Properties

Classification of Real Numbers

The set of real numbers (\mathbb{R}) includes all numbers that can be found on the number line. Real numbers are classified into several subsets:

• Natural Numbers (\mathbb{N}): Counting numbers {1, 2, 3, ...}

• Whole Numbers: Natural numbers plus zero {0, 1, 2, ...}

• Integers (\mathbb{Z}): Whole numbers and their negatives {..., -2, -1, 0, 1, 2, ...}

• Rational Numbers (\mathbb{Q}): Numbers that can be expressed as a fraction of two integers.

• Irrational Numbers: Numbers that cannot be written as fractions (e.g., \sqrt{2}, \pi).

• Real Numbers (\mathbb{R}): All rational and irrational numbers.

Order of Operations

To evaluate mathematical expressions correctly, follow the order of operations (PEMDAS):

• P: Parentheses (grouping symbols)

• E: Exponents and roots

• M/D: Multiplication and Division (left to right)

• A/S: Addition and Subtraction (left to right)

Properties of Real Numbers

Real numbers follow several important properties:

Property	Addition	Multiplication
Commutative	a + b = b + a	ab = ba
Associative	(a + b) + c = a + (b + c)	(ab)c = a(bc)
Identity	a + 0 = a	1(a) = a
Inverse	a + (-a) = 0	a * (1/a) = 1, a ≠ 0
Closure	a + b is real	ab is real
Distributive	a(b + c) = ab + ac

Absolute Value and Inequalities

• Absolute Value: The distance from zero on the number line. Definition: $|a| = \begin{cases} a & a \geq 0 \\ -a & a < 0 \end{cases}$

• Inequalities: Symbols such as <, >, ≤, ≥ are used to compare numbers.

R.3 Exponents and Polynomials

Exponent Rules

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Power of a Product: $(ab)^m = a^m b^m$

• Power of a Quotient: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

• Negative Exponent: $a^{-m} = \frac{1}{a^m}$

• Zero Exponent: $a^0 = 1$ (a ≠ 0)

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

Polynomials: Vocabulary and Operations

• Polynomial: An algebraic expression consisting of terms with non-negative integer exponents.

• Term: A single product of numbers and variables.

• Coefficient: The numerical factor of a term.

• Degree: The highest exponent of the variable in the polynomial.

• Standard (Descending) Order: Terms are written from highest to lowest degree.

• Like Terms: Terms with the same variables and exponents.

Types of Polynomials by Degree:

• Linear: Degree 1

• Quadratic: Degree 2

• Cubic: Degree 3

• Quartic: Degree 4

• Polynomial: Degree greater than 4

Types by Number of Terms: Monomial (1), Binomial (2), Trinomial (3), Polynomial (more than 3)

Operations with Polynomials

• Addition/Subtraction: Combine like terms.

• Multiplication: Use distributive property or FOIL for binomials.

• Special Products:

  • Sum and Difference: $(x + y)(x - y) = x^2 - y^2$

  • Square of a Binomial: $(x + y)^2 = x^2 + 2xy + y^2$

  • $(x - y)^2 = x^2 - 2xy + y^2$

• Division: Use long division or synthetic division for polynomials.

R.4 Factoring Polynomials

Factoring with Greatest Common Factor (GCF)

Always factor out the greatest common factor (GCF) first when factoring polynomials.

• Example: $9y^5 + y^2 = y^2(9y^3 + 1)$

• Example: $6x^2t + 8xt + 12t = 2t(3x^2 + 4x + 6)$

Factoring by Grouping

Factoring by grouping is used for polynomials with four terms. Group terms, factor out the GCF from each group, and factor the common binomial.

Factoring Trinomials

• Type 1: $x^2 + bx + c$

• Type 2: $ax^2 + bx + c$ (use the ac method: multiply a and c, factor, divide off a, and move denominators up)

• Perfect Square Trinomials:

  • $x^2 + 2xy + y^2 = (x + y)^2$

  • $x^2 - 2xy + y^2 = (x - y)^2$

Factoring Special Forms

• Difference of Squares: $x^2 - y^2 = (x - y)(x + y)$

• Sum and Difference of Cubes:

  • $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

  • $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

Factoring by Substitution

For complex expressions, use substitution to simplify the polynomial before factoring.

• Example: $6z^4 - 13z^2 - 5$ (let $u = z^2$)

Factoring is a critical skill that will be used throughout algebra and trigonometry.