P.1 – Algebraic Expressions and Real Numbers

Order of Operations

Order of operations is a fundamental rule for evaluating mathematical expressions. It ensures consistency and accuracy in calculations.

• PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).

• Example: To evaluate , first compute , then , finally .

Sets and Names of Numbers

Numbers are classified into various sets based on their properties.

• Natural Numbers:

• Whole Numbers:

• Integers:

• Rational Numbers: Numbers that can be written as , where and are integers and .

• Irrational Numbers: Numbers that cannot be written as fractions, e.g., , .

• Real Numbers: All rational and irrational numbers.

Inequalities

Inequalities compare the relative size of numbers or expressions.

• Symbols: (less than), (greater than), (less than or equal to), (greater than or equal to).

• Example: means is any number greater than 3.

Absolute Values

The absolute value of a number is its distance from zero on the number line, always non-negative.

• Definition: if , if .

• Example:

Properties of Real Numbers

These properties are essential for simplifying and manipulating algebraic expressions.

• Commutative Property: ,

• Associative Property: ,

• Distributive Property:

• Identity Property: ,

• Inverse Property: , (for )

P.2 – Exponents and Scientific Notation

Exponents

Exponents represent repeated multiplication of a base number.

• Definition: means multiplied by itself times.

• Properties:

  • (for )

• Example:

Common Mistakes

• Incorrect Distribution: ; correct:

• Multiplication:

• Negative Exponents:

• Example:

Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. Note: This section is skipped in the provided materials.

P.3 – Radicals and Rational Exponents

Simplifying Roots

Roots are the inverse operation of exponents. Simplifying roots involves expressing them in their simplest form.

• Square Root: is a number that, when squared, gives .

• Cube Root: is a number that, when cubed, gives .

• Example:

• Example: (since ; $16\sqrt[3]{16}$ is irrational)

Rationalizing Denominators

Expressions are not considered simplified if the denominator contains a radical or an irrational number.

• Rationalizing: Multiply numerator and denominator by a suitable value to eliminate radicals from the denominator.

• Example:

Rational Exponents

Rational exponents are another way to represent roots.

• Definition:

• Example:

• Example:

Converting Between Exponents and Radicals

• Exponent to Radical:

• Radical to Exponent:

• Example:

Simplifying with Rational Exponents

• Example:

• Example:

P.4 – Polynomials

Basics of Polynomials

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

• General Form:

• Degree: The highest power of the variable.

• Example: is a polynomial of degree 2.

Additional info: Detailed study of polynomials will be covered in Chapter 3.

P.5 – Factoring

Factoring Overview

Factoring is rewriting an expression as a product of simpler expressions (factors).

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• Example: ; GCF is , so

• Grouping: Used when terms can be grouped to factor by pairs.

• Example:

• Trinomials: Factor expressions of the form .

• Example:

• Difference of Squares:

• Example:

• Sum/Difference of Cubes: ;

• Example:

P.6 – Rational Expressions

Finding the Domain

The domain of a rational expression is all real numbers except those that make the denominator zero.

• Example: ; domain excludes and .

Simplifying Rational Expressions

Simplifying involves factoring numerators and denominators and canceling common factors.

• Example: (for )

• Example: ; factor numerator and denominator: (for and )

HTML Table: Types of Factoring

The following table summarizes common factoring methods:

Additional info: Factoring is foundational for solving equations, simplifying expressions, and understanding polynomial functions.