Find the 3rd term in the expansion of (3x+4)^3 in simplest form

To find the 3rd term in the expansion of \((3x + 4)^3\), we use the binomial expansion formula:

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

Here, \(a = 3x\), \(b = 4\), and \(n = 3\).

The general term (k+1) in the expansion is given by:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]

For the 3rd term (\(k = 2\)):

\[
T_3 = \binom{3}{2} (3x)^{3-2} (4)^2
\]

Now, let's calculate it step by step:

1. Calculate the binomial coefficient:

\[
\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3
\]

2. Calculate \((3x)^{3-2} = (3x)^1 = 3x\).

3. Calculate \(4^2 = 16\).

Now, substitute these values into the formula for \(T_3\):

\[
T_3 = 3 \cdot 3x \cdot 16 = 144x
\]

Therefore, the 3rd term in the expansion of \((3x + 4)^3\) is \(\mathbf{144x}\).

Do you have any questions or need further details on this?

Here are 5 related questions:
1. What is the 2nd term in the expansion of \((3x + 4)^4\)?
2. How would you find the 4th term in the expansion of \((2x - 5)^3\)?
3. Can you expand \((2x + 1)^3\) using the binomial theorem?
4. What is the sum of all the coefficients in the expansion of \((3x + 4)^3\)?
5. How does the sign change in the expansion of \((3x - 4)^3\)?

**Tip:** When expanding binomials, always identify the general term formula first to easily find any specific term in the expansion.

Learn how to find the 3rd term in the expansion of (3x + 4)^3 with a step-by-step solution. This problem involves using the binomial theorem and calculating binomial coefficients, suitable for advanced high school students.

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