Math Problem Statement
Solution
The image shows a mathematical problem where we are asked to find the term free from in the expansion of the binomial expression:
To solve this, we will expand the binomial expression using the binomial theorem, which is given by:
In this case, , , and .
The general term in the expansion of is:
Simplifying this expression:
This further simplifies to:
Now, we are asked to find the term free from . For a term to be independent of , the exponent of must be zero. Therefore, we set the exponent of in the general term to zero:
Solving for :
So, the term free from occurs when . Substituting into the general term formula:
Since , we have:
Now, calculate , which is equal to because of symmetry:
Thus, the term free from is:
Finally, calculate :
Therefore, the term free from is:
So, the constant term is .
Would you like further details on this solution?
Here are 5 related questions:
- How does the binomial theorem apply to polynomial expansions?
- What is the role of the general term in identifying specific terms in an expansion?
- Why is the exponent of set to zero to find the constant term?
- How can you compute binomial coefficients using Pascal’s triangle?
- Can the same process be applied to other powers or terms in binomial expansions?
Tip: Binomial expansions can often simplify complex algebraic expressions and are particularly useful when looking for specific terms or behaviors of a function.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Binomial Theorem
Exponent Rules
Formulas
Binomial expansion formula: (a + b)^n = Σ binom(n, k) * a^(n-k) * b^k
Exponent rules for simplifying expressions with powers of variables
Theorems
Binomial Theorem
Suitable Grade Level
Grades 10-12
Related Recommendation
Find the Term Free from x in the Expansion of (x^2 - 6/x)^9
Constant Term of (x - 2x^-2)^9 Using Binomial Expansion
Find the Term Independent of x in the Expansion of (1/x)(1/3x - x/2)^9
Find the Constant Term in the Expansion of \( \frac{1}{x^3} \left( \frac{1}{3x^2} - \frac{x}{2} \right)^9 \)
Finding the Independent Term in the Expansion of (x - 1/x^2)^9