Methods for Factoring Polynomials

Study Notes

Factoring Methods for Polynomials

Greatest Common Factor (GCF): Identify and factor out the highest common multiple of all polynomial terms. Always apply this method first.
Grouping: Utilized for polynomials with four terms, involving two steps:
- Step 1: Group terms to form two pairs. For example, (3xy + 3y + 2ax + 2a) becomes ((3xy + 3y) + (2ax + 2a)).
- Step 2: Factor out common elements from each group, forming binomials: ((x+1)(3y+2a)). Verify results using the FOIL method, paying attention to signs.
Trinomial Factoring: Reverses the FOIL method to find two binomials. Required that the middle term equals the sum of the outer and inner products: (a^2 - b + 1) factors to ((a+b)(a+b)).
Difference of Squares: Applies when given two terms in the form (a^2 - b^2), factoring to ((a+b)(a-b)). Ensure it involves two terms being subtracted.
Sum of Squares: Cannot be factored further as (a^2 + b^2) is prime. Identifies polynomials represented by two added terms.
Difference of Cubes: Utilizes the formula (a^3 - b^3 = (a-b)(a^2 + ab + b^2)). Requires both terms in the polynomial to be cubes and be in a binomial format.
Sum of Cubes: Factors as (a^3 + b^3 = (a+b)(a^2 - ab + b^2)). Recognizes that both terms are cubes and emphasizes the need for sign changes in the middle term.

Fundamental Properties

Associative Property: States that the grouping of numbers does not affect the result in addition or multiplication; the placement of parentheses is inconsequential.
Commutative Property: Order of numbers in addition or multiplication does not change the product; the sequence of terms can be rearranged freely.
Distributive Property: Explains multiplication of a sum by distributing over each term separately and then combining the products; frequently used in FOIL expansions.

Mathematical Definitions

Prime Expression: An expression that cannot be factored into simpler expressions; has no prime factors.
Factor Completely: The process of writing an expression as a product of prime factors only.

Common Cubes and Squares

Common Cubes (n³):
- Values include 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000 for integers 1 through 10.
Common Squares (n²):
- Values include 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 for integers 1 through 10.

Additional Concepts

Conjugate: Two similar expressions or quantities that differ only by the sign in the middle, significant in simplification processes.
Tips for Difference of Squares:
- Having a positive and negative term leads to cancellation of the middle term.
- If both terms are even, 2 is a common factor, aiding in simplification.

Description

This quiz covers seven essential methods for factoring polynomials, focusing on the greatest common factor and grouping. Each method is explained with definitions and examples for clarity. Perfect for students looking to master polynomial factorization.