https://file.aiquickdraw.com/mathbot/file/1727296699045gi7u14x5.jpg

The function to expand is:

\[
f(x) = (x + 7)^3
\]

This expression represents a binomial raised to the third power. To expand it, we can use the **Binomial Theorem**, or we can directly expand using multiplication.

\[
f(x) = (x + 7)^3 = (x + 7)(x + 7)(x + 7)
\]

First, we will expand two of the binomials:

\[
(x + 7)(x + 7) = x^2 + 7x + 7x + 49 = x^2 + 14x + 49
\]

Now multiply the result by the remaining \((x + 7)\):

\[
(x^2 + 14x + 49)(x + 7) = x^2(x + 7) + 14x(x + 7) + 49(x + 7)
\]

Expanding each term:

\[
x^2(x + 7) = x^3 + 7x^2
\]
\[
14x(x + 7) = 14x^2 + 98x
\]
\[
49(x + 7) = 49x + 343
\]

Now, combine all the terms:

\[
x^3 + 7x^2 + 14x^2 + 98x + 49x + 343 = x^3 + (7x^2 + 14x^2) + (98x + 49x) + 343
\]

Simplifying:

\[
f(x) = x^3 + 21x^2 + 147x + 343
\]

Thus, the expanded function is:

\[
f(x) = x^3 + 21x^2 + 147x + 343
\]

Would you like more details or have any further questions?

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Here are 5 related questions:
1. How would you expand \((x + 5)^4\)?
2. Can you apply the Binomial Theorem to expand \((x - 3)^5\)?
3. How does the distributive property help in expanding binomials?
4. What are the key steps in expanding \((2x + 4)^3\)?
5. How does the expansion of \((x + a)^n\) change as \(n\) increases?

**Tip**: Memorizing the expansions of small powers of binomials can help you quickly solve similar problems.

Learn how to expand the polynomial function f(x) = (x + 7)^3 using the Binomial Theorem and direct multiplication. This guide walks you through key algebraic steps suitable for students in Grades 10-12.

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