In Exercises 35–54, use the FOIL method to multiply the binomials. (3xⁿ−yⁿ)(xⁿ+2yⁿ)

Verified step by step guidance

1. Identify the binomials to be multiplied: \((3x^n - y^n)\) and \((x^n + 2y^n)\).
2. Apply the FOIL method, which stands for First, Outer, Inner, Last, to multiply the binomials.
3. First: Multiply the first terms of each binomial: \(3x^n \cdot x^n\).
4. Outer: Multiply the outer terms of the binomials: \(3x^n \cdot 2y^n\).
5. Inner: Multiply the inner terms of the binomials: \(-y^n \cdot x^n\).

Key Concepts

FOIL Method

The FOIL method is a technique used to multiply two binomials. It stands for First, Outside, Inside, Last, referring to the order in which you multiply the terms. This method helps ensure that all combinations of the terms are accounted for, leading to the correct expansion of the expression.

Binomials

A binomial is a polynomial that consists of exactly two terms, which can be separated by a plus or minus sign. In the expression (3xⁿ−yⁿ)(xⁿ+2yⁿ), both factors are binomials. Understanding how to manipulate binomials is essential for performing operations like multiplication and factoring in algebra.

Polynomial Expansion

Polynomial expansion involves distributing each term in one polynomial to every term in another polynomial. This process results in a new polynomial that combines like terms. Mastery of polynomial expansion is crucial for simplifying expressions and solving equations in algebra.