Use the binomial series to expand the function as a power series.
7
(4 + x)3

To expand the function \(7(4 + x)^3\) as a power series using the binomial theorem, let's proceed step by step.

### Step 1: Recognize the binomial theorem formula
The binomial theorem states that:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
where \(\binom{n}{k}\) is the binomial coefficient, given by:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

### Step 2: Apply the binomial theorem to \((4 + x)^3\)
Here, \(a = 4\), \(b = x\), and \(n = 3\). So, applying the binomial theorem:
\[
(4 + x)^3 = \sum_{k=0}^{3} \binom{3}{k} 4^{3-k} x^k
\]

### Step 3: Expand the series
Now, we calculate each term of the series for \(k = 0\) to \(k = 3\):

- For \(k = 0\):
\[
\binom{3}{0} 4^{3-0} x^0 = 1 \times 4^3 \times 1 = 64
\]

- For \(k = 1\):
\[
\binom{3}{1} 4^{3-1} x^1 = 3 \times 4^2 \times x = 3 \times 16 \times x = 48x
\]

- For \(k = 2\):
\[
\binom{3}{2} 4^{3-2} x^2 = 3 \times 4^1 \times x^2 = 3 \times 4 \times x^2 = 12x^2
\]

- For \(k = 3\):
\[
\binom{3}{3} 4^{3-3} x^3 = 1 \times 4^0 \times x^3 = 1 \times 1 \times x^3 = x^3
\]

So, the expansion is:
\[
(4 + x)^3 = 64 + 48x + 12x^2 + x^3
\]

### Step 4: Multiply by 7
Finally, multiply the entire expansion by 7:
\[
7(4 + x)^3 = 7(64 + 48x + 12x^2 + x^3)
\]
Distributing the 7 gives:
\[
7(4 + x)^3 = 448 + 336x + 84x^2 + 7x^3
\]

### Final Answer:
The power series expansion of \(7(4 + x)^3\) is:
\[
7(4 + x)^3 = 448 + 336x + 84x^2 + 7x^3
\]

Would you like any further details or explanations? Here are some related questions you might find useful:

1. How do you calculate binomial coefficients?
2. What is the binomial theorem, and how does it work?
3. How can you use the binomial theorem for negative or fractional exponents?
4. What are some practical applications of the binomial theorem?
5. How does the binomial theorem relate to Pascal's triangle?
6. Can the binomial theorem be used for series expansions involving more than two terms?
7. How do you find the general term in a binomial expansion?
8. How can you simplify binomial expansions for large powers?

**Tip:** The binomial theorem is especially powerful for expanding expressions with large exponents, saving time compared to direct multiplication.

Learn how to expand 7(4 + x)^3 into a power series using the binomial theorem with detailed step-by-step instructions. This problem covers key mathematical concepts, including the binomial theorem and binomial coefficients, suitable for advanced high school students.

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