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The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions that are raised to a power. The theorem states: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k \]. Here, \(a\) and \(b\) are any numbers, and \(n\) is a non-negative integer. This formula gives us a way to calculate each term in the expansion. It uses binomial coefficients \( \binom{n}{k} \), which we will cover in the next section. The Binomial Theorem can simplify complex polynomial expansions. For instance, expanding \((b - 5)^4\) utilizes the same principles, where \(a = b\), \(b = -5\), and \(n = 4\). It breaks down complex multiplication into manageable terms.

Binomial coefficients \( \binom{n}{k} \) are key components in the Binomial Theorem. They are calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]. The exclamation mark denotes factorial, which is the product of all positive integers up to that number. For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). For \((b - 5)^4\), we compute the coefficients as follows:

• \( \binom{4}{0} = 1 \)
• \( \binom{4}{1} = 4 \)
• \( \binom{4}{2} = 6 \)
• \( \binom{4}{3} = 4 \)
• \( \binom{4}{4} = 1 \)

These coefficients determine how many ways we can choose \(k\) elements from \(n\) elements, which helps us in placing the respective powers of \(a\) and \(b\) in the expansion.

Polynomial expansion is the process of expressing a polynomial that is raised to a power as a sum of simpler terms. The Binomial Theorem provides a systematic method to achieve this. To expand \((b - 5)^4\), follow these steps:

• Identify \(a = b\), \(b = -5\), and \(n = 4\).
• Use the binomial coefficients \( \binom{4}{k} \) for \(k = 0, 1, 2, 3, 4\).
• Calculate each term using \( \binom{4}{k} b^{4-k} (-5)^k \).
• Combine the terms.

For example, calculating each term for \((b - 5)^4\): \[ b^4 - 20b^3 + 150b^2 - 500b + 625\] First term: \( 1 \times b^4 \times 1 = b^4 \)
Second term: \( 4 \times b^3 \times (-5) = -20b^3 \)
Third term: \( 6 \times b^2 \times 25 = 150b^2 \)
Fourth term: \( 4 \times b \times (-125) = -500b \)
Fifth term: \( 1 \times 1 \times 625 = 625 \) This step-by-step approach simplifies the polynomial expansion process, making it easier to understand and apply in other problems.