https://file.aiquickdraw.com/mathbot/file/17265667693676hkvyfbq.jpg

The image contains two mathematical problems related to **binomial expansions** and finding the coefficients of specific terms. Here's an interpretation of the problems:

1. **Problem 1:**
   - The expression \(\left( 2x^2 - \frac{1}{2x} \right)^6\) is expanded, and the coefficient of the term \(x^3\) is requested.
   
   The solution uses the general term from the binomial expansion formula:
   \[
   \binom{6}{r} \left(2x^2\right)^{6-r} \left(-\frac{1}{2x}\right)^r
   \]
   We need to find the value of \(r\) that gives the \(x^3\) term. The powers of \(x\) are calculated, and then the value of \(r\) is determined, leading to the coefficient of \(x^3\).

2. **Problem 2:**
   - The expression \(\left(1 + 3x - x^2\right)^8\) is expanded, and the coefficient of \(x^3\) is requested.
   
   The general term for this expansion is:
   \[
   \binom{8}{p, q, r} \cdot 1^p \cdot (3x)^q \cdot (-x^2)^r
   \]
   where \(p + q + r = 8\) and the sum of powers for \(x\) should be \(3\). The combinations of \(p, q, r\) that satisfy this condition are calculated, and the coefficient is obtained.

Would you like me to walk through each solution in more detail or explain a specific part further? 

---

Here are 5 related questions to deepen your understanding:
1. How do you derive the general term in the binomial expansion of \((a + b)^n\)?
2. What is the significance of binomial coefficients in expansions?
3. How can you apply the multinomial theorem for three variables like in problem 2?
4. What conditions must be met for the powers of \(x\) in a multinomial expansion?
5. How would the solution change if the powers or terms inside the binomial were different?

**Tip**: When dealing with binomial expansions, always carefully track the powers of the variables to find the correct term and coefficient.

Find the coefficient of the x^3 term in the expansion of (1 + 3x - x^2)^8.

Explore the binomial and multinomial expansion concepts through the detailed step-by-step solution to finding the x^3 term coefficient in the expansion of (1 + 3x - x^2)^8. Suitable for university-level mathematics 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