AP Precalculus Practice Test: Unit 1 Question #31 Find the Third Term of (2x+3)^5 Binomial Theorem

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To find the third term of the expansion of \( (2x + 3)^5 \) using the **Binomial Theorem**, we apply the formula:

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

In this case:
- \( a = 2x \)
- \( b = 3 \)
- \( n = 5 \)

We are looking for the **third term**, which corresponds to \( k = 2 \) because the term index starts from \( k = 0 \).

### Step 1: Use the Binomial Theorem Formula for \( k = 2 \)
The general term in the expansion is:

\[
T_k = \binom{n}{k} a^{n-k} b^k
\]

For \( k = 2 \), we have:

\[
T_2 = \binom{5}{2} (2x)^{5-2} 3^2
\]

### Step 2: Calculate the Individual Components
- \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \)
- \( (2x)^{5-2} = (2x)^3 = 8x^3 \)
- \( 3^2 = 9 \)

### Step 3: Put Everything Together

\[
T_2 = 10 \cdot 8x^3 \cdot 9
\]

Simplifying:

\[
T_2 = 10 \cdot 72x^3 = 720x^3
\]

### Final Answer:
The third term of \( (2x + 3)^5 \) is \( \boxed{720x^3} \).

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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