Use the Binomial Theorem to expand each binomial and express the result in simplified form. (c+2)⁵

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Recall the Binomial Theorem formula for expanding \((a + b)^n\): \[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \] where \(\binom{n}{k}\) is the binomial coefficient calculated as \(\frac{n!}{k!(n-k)!}\).

Identify the values in the given expression \((c + 2)^5\): here, \(a = c\), \(b = 2\), and \(n = 5\).

Write out the expansion terms using the formula: \[ \sum_{k=0}^5 \binom{5}{k} c^{5-k} (2)^k = \binom{5}{0} c^5 (2)^0 + \binom{5}{1} c^4 (2)^1 + \binom{5}{2} c^3 (2)^2 + \binom{5}{3} c^2 (2)^3 + \binom{5}{4} c^1 (2)^4 + \binom{5}{5} c^0 (2)^5 \]

Calculate each binomial coefficient \(\binom{5}{k}\) for \(k=0\) to \(5\) and write each term explicitly with powers of \(c\) and \(2\).

Simplify each term by evaluating the powers of 2 and multiply by the binomial coefficients, then combine all terms to write the final expanded polynomial.

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주요 개념

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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 equals the sum of terms involving binomial coefficients multiplied by powers of a and b. This theorem simplifies the expansion process without multiplying the binomial repeatedly.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from a set of n elements. They appear as coefficients in the binomial expansion and can be found using Pascal's Triangle or the formula C(n, k) = n! / (k!(n-k)!). These coefficients determine the weight of each term in the expansion.

Exponent Rules in Expansion

When expanding (a + b)^n, each term involves powers of a and b such that the exponents add up to n. Specifically, the k-th term has a raised to the power (n-k) and b raised to the power k. Understanding how to apply exponent rules ensures correct simplification of each term in the expanded expression.

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (c+2)5