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The image contains problems related to the binomial expansion. Specifically, you need to find the first three terms in the expansion of the following expressions:

1. \((a + b)^8\)
2. \((x^2 + 3)^5\)
3. \((x + 3y)^9\)

Let's solve each one using the binomial theorem.

### 1. Expansion of \((a + b)^8\)
The binomial expansion for \((a + b)^8\) is given by:
\[
(a + b)^8 = \sum_{k=0}^{8} \binom{8}{k} a^{8-k} b^k
\]
The first three terms are:
\[
\text{1st term: } \binom{8}{0} a^8 b^0 = a^8
\]
\[
\text{2nd term: } \binom{8}{1} a^7 b^1 = 8a^7b
\]
\[
\text{3rd term: } \binom{8}{2} a^6 b^2 = 28a^6b^2
\]
So, the first three terms are \( a^8 + 8a^7b + 28a^6b^2 \).

### 2. Expansion of \((x^2 + 3)^5\)
The binomial expansion for \((x^2 + 3)^5\) is given by:
\[
(x^2 + 3)^5 = \sum_{k=0}^{5} \binom{5}{k} (x^2)^{5-k} 3^k
\]
The first three terms are:
\[
\text{1st term: } \binom{5}{0} (x^2)^5 3^0 = x^{10}
\]
\[
\text{2nd term: } \binom{5}{1} (x^2)^4 3^1 = 15x^8 \cdot 3 = 15x^8 \cdot 3 = 15x^8 \cdot 3 = 15x^8
\]
\[
\text{3rd term: } \binom{5}{2} (x^2)^3 3^2 = 10x^6 \cdot 9 = 90x^6
\]
So, the first three terms are \( x^{10} + 15x^8 + 90x^6 \).

### 3. Expansion of \((x + 3y)^9\)
The binomial expansion for \((x + 3y)^9\) is given by:
\[
(x + 3y)^9 = \sum_{k=0}^{9} \binom{9}{k} x^{9-k} (3y)^k
\]
The first three terms are:
\[
\text{1st term: } \binom{9}{0} x^9 (3y)^0 = x^9
\]
\[
\text{2nd term: } \binom{9}{1} x^8 (3y)^1 = 9x^8 \cdot 3y = 27x^8y
\]
\[
\text{3rd term: } \binom{9}{2} x^7 (3y)^2 = 36x^7 \cdot 9y^2 = 324x^7y^2
\]
So, the first three terms are \( x^9 + 27x^8y + 324x^7y^2 \).

Would you like more details on any part of this, or do you have any questions?

Here are 5 related questions you might consider:
1. How do you derive the binomial coefficient \(\binom{n}{k}\)?
2. What is the general term in the binomial expansion of \((a + b)^n\)?
3. How does the binomial expansion change if one of the terms is negative, e.g., \((a - b)^n\)?
4. How can the binomial theorem be applied to approximate functions using polynomials?
5. How would you find the middle term(s) in a binomial expansion?

**Tip:** When expanding a binomial expression, it's helpful to recognize that the coefficients in the expansion correspond to the entries in Pascal's Triangle.

Learn how to expand the binomial expression (x + 3y)^9 using the binomial theorem with a detailed step-by-step solution. This problem covers binomial coefficients, the binomial expansion formula, and the application of the binomial theorem in algebra. Suitable for high school students.

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