Use the Binomial Theorem to expand each binomial and express the result in simplified form. $(5x\text{–}1{)}^3$

Verified step by step guidance

1. Identify the binomial expression to expand: \( (5x - 1)^3 \). Here, the binomial has terms \(a = 5x\) and \(b = -1\), and the exponent \(n = 3\).
2. Recall the Binomial Theorem formula for expansion: \( (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \), where \(\binom{n}{k}\) is the binomial coefficient.
3. Write out each term of the expansion using the formula: \( \binom{3}{0} (5x)^3 (-1)^0 + \binom{3}{1} (5x)^2 (-1)^1 + \binom{3}{2} (5x)^1 (-1)^2 + \binom{3}{3} (5x)^0 (-1)^3 \).
4. Calculate each binomial coefficient: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).
5. Simplify each term by raising \$5x\( and \)-1$ to the appropriate powers, multiply by the binomial coefficients, and then combine all terms to write the expanded polynomial.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions raised to a power, such as (a + b)^n. It uses binomial coefficients, represented by combinations, to determine the coefficients of each term in the expansion. This theorem simplifies the process of expanding powers of binomials without direct multiplication.

Binomial Coefficients and Combinations

Binomial coefficients are the numerical factors in the expansion and correspond to combinations, denoted as n choose k (C(n, k)). They represent the number of ways to select k elements from n and are calculated using factorials. These coefficients determine the weight of each term in the binomial expansion.

Simplification of Polynomial Expressions

After expanding the binomial using the theorem, the resulting polynomial must be simplified by combining like terms and performing arithmetic operations. This step ensures the final expression is in its simplest form, making it easier to interpret and use in further calculations.