what is the coefficient of x^2 in the expansion of (2+x)^5

To determine the coefficient of \( x^2 \) in the expansion of \( (2 + x)^5 \), we can use the binomial theorem. The binomial theorem states:

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

where \( \binom{n}{k} \) represents the binomial coefficient.

In this problem:
- \( a = 2 \),
- \( b = x \),
- \( n = 5 \).

We are interested in the term involving \( x^2 \). For this, \( k = 2 \), since the exponent of \( x \) is \( k \).

The corresponding term in the expansion is:

\[
\binom{5}{2} \cdot 2^{5-2} \cdot x^2.
\]

Calculate the coefficient:

\[
\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10,
\]

and

\[
2^{5-2} = 2^3 = 8.
\]

Therefore, the coefficient of \( x^2 \) is:

\[
10 \times 8 = 80.
\]

So, the coefficient of \( x^2 \) in the expansion of \( (2 + x)^5 \) is **80**.

Would you like more details or have any other questions? Here are five related questions to explore:

1. What is the coefficient of \( x^3 \) in \( (2 + x)^5 \)?
2. How do you expand \( (a + b)^n \) using the binomial theorem?
3. What is the general form of a binomial expansion?
4. Can you determine the coefficient of \( x^4 \) in \( (3 + x)^6 \)?
5. How would the expansion change if \( x \) had a negative coefficient?

**Tip**: In binomial expansions, always identify the value of \( k \) corresponding to the power of \( x \) you need to determine the coefficient.

Learn how to find the coefficient of x^2 in the expansion of (2 + x)^5 using the binomial theorem and binomial coefficients. This detailed explanation covers the application of binomial theorem, calculation of binomial coefficients, and step-by-step process to determine the coefficient. Suitable for high school students and beyond.

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