Math Problem Statement
Solution
The given image shows a question asking for the coefficients of and in the expansion of .
We will use the Binomial Theorem to expand this expression. The Binomial Theorem states:
For the given expression :
The general term in the expansion is:
This term simplifies to:
Finding the coefficient of :
We need to find the term where the power of is 2. That means , so . Substituting into the general term:
Simplifying:
Since , the coefficient of is:
Finding the coefficient of :
We need to find the term where the power of is 3. That means , so . Substituting into the general term:
Simplifying:
Since , the coefficient of is:
Final Answers:
- The coefficient of is 1740.
- The coefficient of is -32,480.
Would you like further details or have any questions about this expansion?
Here are five related questions to explore:
- How do binomial coefficients change when the exponent increases?
- Can you find the coefficient of in the expansion of ?
- What is the sum of all the coefficients in the expansion of ?
- How would the expansion differ if the expression were instead?
- What is the general form of the binomial theorem for negative exponents?
Tip: The binomial coefficients can be quickly calculated using Pascal's triangle or the formula for combinations .
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Math Problem Analysis
Mathematical Concepts
Binomial Expansion
Polynomial Coefficients
Combinatorics
Formulas
Binomial Expansion: (a + b)^n = ∑(n choose k) * a^(n-k) * b^k
Combination Formula: (n choose k) = n! / [k!(n - k)!]
Theorems
Binomial Theorem
Suitable Grade Level
Grades 11-12
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