To expand the binomial \((3x^2 + 1)^3\) using the Binomial Theorem, we'll apply the formula:

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

Here, \(a = 3x^2\), \(b = 1\), and \(n = 3\).

Let's calculate each term:

1. **Term 1**: \(k = 0\)  
   \[
   \binom{3}{0} (3x^2)^{3} (1)^{0} = 1 \times (3x^2)^3 = 27x^6
   \]

2. **Term 2**: \(k = 1\)  
   \[
   \binom{3}{1} (3x^2)^{2} (1)^1 = 3 \times (3x^2)^2 = 27x^4
   \]

3. **Term 3**: \(k = 2\)  
   \[
   \binom{3}{2} (3x^2)^{1} (1)^2 = 3 \times (3x^2) = 9x^2
   \]

4. **Term 4**: \(k = 3\)  
   \[
   \binom{3}{3} (3x^2)^{0} (1)^3 = 1 \times 1 = 1
   \]

Now, we combine all these terms:

\[
(3x^2 + 1)^3 = 27x^6 + 27x^4 + 9x^2 + 1
\]

### Do you want more details or have any questions? 

Here are some related questions:
1. How do you expand a binomial using the Binomial Theorem?
2. What are binomial coefficients, and how are they calculated?
3. How would you expand \((2x + 3)^4\) using the Binomial Theorem?
4. What is Pascal’s Triangle, and how does it relate to binomial expansions?
5. How do you find the middle term in a binomial expansion?

**Tip:** Remember that the sum of the exponents in each term of a binomial expansion equals the power \(n\) of the binomial.

Learn how to expand the binomial (3x^2 + 1)^3 using the Binomial Theorem with a step-by-step solution. This problem covers key concepts such as binomial expansion, combinatorics, and algebraic manipulation. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation