Precalculus: Algebraic Expressions, Factoring, and Polynomial Expansion

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 $(a+b)^2$ yields $a^2 + 2ab + b^2$.

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 $(a+b)^n$.

• Binomial Expansion: The process of expanding expressions like $(a+b)^n$ using the distributive property or binomial theorem.

• Common Expansions:

  • $(a+b)^2 = a^2 + 2ab + b^2$

  • $(a-b)^2 = a^2 - 2ab + b^2$

  • $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

  • $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

  • $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$

Example: Expand $(x+y)^3$:

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

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: $a^2 - b^2 = (a-b)(a+b)$

• Perfect Square Trinomial: $a^2 + 2ab + b^2 = (a+b)^2$

• Sum and Difference of Cubes:

  • $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

  • $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

Example: Factor $x^2 - 9$:

$x^2 - 9 = (x-3)(x+3)$

Summary Table: Common Algebraic Expansions

Expression	Expanded Form
$(a+b)^2$	$a^2 + 2ab + b^2$
$(a-b)^2$	$a^2 - 2ab + b^2$
$(a+b)^3$	$a^3 + 3a^2b + 3ab^2 + b^3$
$(a-b)^3$	$a^3 - 3a^2b + 3ab^2 - b^3$
$(a+b+c)^2$	$a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$

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