In Exercises 15–58, find each product. (4x²−1)²

Verified step by step guidance

1. Recognize that the expression \((4x^2 - 1)^2\) is a binomial squared. Use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\).
2. Identify \(a = 4x^2\) and \(b = 1\) in the given expression.
3. Apply the formula: \((4x^2 - 1)^2 = (4x^2)^2 - 2(4x^2)(1) + (1)^2\).
4. Simplify each term: \((4x^2)^2 = 16x^4\), \(-2(4x^2)(1) = -8x^2\), and \((1)^2 = 1\).
5. Combine the simplified terms to express the expanded form: \(16x^4 - 8x^2 + 1\).

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. This concept is essential for simplifying expressions like (4x^2 - 1)^2.

Square of a Binomial

The square of a binomial, expressed as (a - b)^2, can be simplified using the formula a^2 - 2ab + b^2. In the context of the given expression (4x^2 - 1)^2, this means squaring the first term, subtracting twice the product of the two terms, and adding the square of the second term. Understanding this formula is crucial for correctly expanding the expression.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials together, which can be done using the distributive property or the FOIL method for binomials. In the case of (4x^2 - 1)^2, you will multiply the binomial by itself, ensuring that each term in the first binomial is multiplied by each term in the second. Mastery of this concept is necessary for accurately calculating the product.