Use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x³ − 1)⁴

Verified step by step guidance

1. Identify the binomial expression to be expanded: \((2x^{3} - 1)^4\).
2. Recall the Binomial Theorem formula: \(\displaystyle (a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k\), where \(\binom{n}{k}\) is the binomial coefficient.
3. Set \(a = 2x^{3}\), \(b = 1\), and \(n = 4\). Write the expansion as \(\sum_{k=0}^{4} \binom{4}{k} (2x^{3})^{4-k} (-1)^k\).
4. Calculate each term by evaluating the binomial coefficient \(\binom{4}{k}\), raising \$2x^{3}\( to the power \)(4-k)\(, and multiplying by \)(-1)^k\( for \)k = 0, 1, 2, 3, 4$.
5. Simplify each term by applying the exponent to \$2\( and \)x^{3}$ separately, then combine like terms to write the final expanded polynomial.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, summing k from 0 to n. This theorem simplifies the expansion process without multiplying the binomial repeatedly.

Binomial Coefficients

Binomial coefficients, denoted as (n choose k), represent the number of ways to choose k elements from n and are calculated using factorials: n! / (k!(n-k)!). They serve as the coefficients in the expanded form of a binomial expression and determine the weight of each term.

Exponent Rules

Exponent rules govern how to handle powers in algebraic expressions, such as multiplying powers with the same base by adding exponents and raising a power to another power by multiplying exponents. These rules are essential when expanding terms like (2x³)^m to simplify the expression correctly.