In Exercises 15–58, find each product. (4x^2−1)^2

Key Concepts

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion can be achieved using the Binomial Theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. In this case, (4x^2 - 1)^2 is a binomial expression that can be expanded using this theorem.

Squaring a Binomial

Squaring a binomial involves applying the formula (a - b)^2 = a^2 - 2ab + b^2. This formula allows us to find the square of a binomial expression by calculating the square of the first term, subtracting twice the product of the two terms, and adding the square of the second term. For (4x^2 - 1)^2, we will identify a as 4x^2 and b as 1 to perform the squaring.

Polynomial Multiplication

Polynomial multiplication is the process of multiplying two polynomials together, which involves distributing each term in the first polynomial to every term in the second polynomial. This results in a new polynomial that combines like terms. In the case of (4x^2 - 1)^2, we will multiply (4x^2 - 1) by itself, ensuring to combine any like terms in the final expression.