Round #1012

Expand (x + 2)^4 using the binomial theorem.

## Step 1: Recall the Binomial Theorem Formula The binomial theorem formula is given by: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\), where \(\binom{n}{k}\) represents the binomial coefficient, which is calculated as \(n! / (k!(n-k)!)\). ## Step 2: Apply the Binomial Theorem to the Given Problem In this problem, \(a = x\), \(b = 2\), and \(n = 4\). We will substitute these values into the binomial theorem formula. ## Step 3: Calculate Each Term We calculate each term for \(k = 0\) to \(k = 4\): - For \(k = 0\), \(\binom{4}{0} x^{4-0}2^0 = 1 \cdot x^4 \cdot 1 = x^4\) - For \(k = 1\), \(\binom{4}{1} x^{4-1}2^1 = 4 \cdot x^3 \cdot 2 = 8x^3\) - For \(k = 2\), \(\binom{4}{2} x^{4-2}2^2 = 6 \cdot x^2 \cdot 4 = 24x^2\) - For \(k = 3\), \(\binom{4}{3} x^{4-3}2^3 = 4 \cdot x^1 \cdot 8 = 32x\) - For \(k = 4\), \(\binom{4}{4} x^{4-4}2^4 = 1 \cdot x^0 \cdot 16 = 16\) ## Step 4: Combine the Terms Now, we combine the calculated terms to get the expanded form of \((x + 2)^4\): \[x^4 + 8x^3 + 24x^2 + 32x + 16\] The final answer is: $\boxed{x^4 + 8x^3 + 24x^2 + 32x + 16}$

```csharp using System; public class BinomialExpansion { public static void Main() { int n = 4; double a = 1; // coefficient of x double b = 2; // constant term Console.WriteLine(ExpandBinomial(n, a, b)); } public static string ExpandBinomial(int n, double a, double b) { string expansion = ""; for (int k = 0; k <= n; k++) { double coefficient = BinomialCoefficient(n, k); double term = coefficient * Math.Pow(a, n - k) * Math.Pow(b, k); if (term != 0) { if (expansion != "") { if (term > 0) { expansion += " + "; } else { expansion += " - "; } } if (Math.Abs(term) != 1 || (n - k == 0 && k == 0)) { expansion += Math.Abs(term); } if (n - k > 0) { expansion += "x"; if (n - k > 1) { expansion += "^" + (n - k); } } if (k > 0) { expansion += (k > 1 ? "^" + k : "") + "·2"; } } } return expansion; } public static double BinomialCoefficient(int n, int k) { if (k > n - k) { k = n - k; } double result = 1; for (int i = 0; i < k; i++) { result = result * (n - i) / (i + 1); } return result; } } ```