To expand \((2 + x)^5\), we can use the **Binomial Theorem**. According to the Binomial Theorem:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case, \(a = 2\), \(b = x\), and \(n = 5\). We calculate each term of the expansion as follows:

\[
(2 + x)^5 = \sum_{k=0}^{5} \binom{5}{k} 2^{5-k} x^k
\]

Now, let's compute each term:

1. \(k = 0\): \(\binom{5}{0} 2^{5} x^0 = 1 \cdot 32 \cdot 1 = 32\)
2. \(k = 1\): \(\binom{5}{1} 2^{4} x^1 = 5 \cdot 16 \cdot x = 80x\)
3. \(k = 2\): \(\binom{5}{2} 2^{3} x^2 = 10 \cdot 8 \cdot x^2 = 80x^2\)
4. \(k = 3\): \(\binom{5}{3} 2^{2} x^3 = 10 \cdot 4 \cdot x^3 = 40x^3\)
5. \(k = 4\): \(\binom{5}{4} 2^{1} x^4 = 5 \cdot 2 \cdot x^4 = 10x^4\)
6. \(k = 5\): \(\binom{5}{5} 2^{0} x^5 = 1 \cdot 1 \cdot x^5 = x^5\)

Now, summing these terms together:

\[
(2 + x)^5 = 32 + 80x + 80x^2 + 40x^3 + 10x^4 + x^5
\]

Thus, the expanded form is:

\[
(2 + x)^5 = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32
\]

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Learn how to expand (2 + x)^5 using the Binomial Theorem. This step-by-step guide walks through the application of the binomial expansion formula, calculating each term individually. Suitable for high school students in Grades 10-12.

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