Fundamental Concepts of Algebra

Algebraic Expressions and Operations

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Understanding how to manipulate these expressions is foundational in precalculus.

• Development/Expansion: The process of expanding products of sums into a sum of terms, often using distributive properties.

• Factoring: Writing an expression as a product of its factors. This is the reverse process of expansion.

• Simplification: Reducing an expression to its simplest form by combining like terms and performing indicated operations.

Example: Expanding yields .

Polynomial Functions

Expansion of Binomials and Polynomials

Expanding binomials and polynomials is a key skill in algebra. The binomial theorem provides a systematic way to expand expressions of the form .

• Binomial Expansion: The process of expanding expressions like using the distributive property or binomial theorem.

• Common Expansions:

Example: Expand :

$

Factoring and Simplification

Factoring Common Patterns

Factoring is the process of expressing a polynomial as a product of its factors. Recognizing common patterns helps in quick factorization.

• Difference of Squares:

• Perfect Square Trinomial:

• Sum and Difference of Cubes:

Example: Factor :

Summary Table: Common Algebraic Expansions

Additional info: These notes cover foundational algebraic manipulations essential for success in precalculus, including expansion, factoring, and simplification of polynomial expressions.