Help to solve binominal theorem

  • MHB
  • Thread starter pusekatja
  • Start date
  • Tags
    Theorem
In summary, the conversation discusses a question about binomial theorem and finding the value of k in a given expansion. The binomial theorem is mentioned and a possible approach is suggested. The conversation ends with a question about the value of i for the term to be a constant.
  • #1
pusekatja
1
0
Hi!
I am struggling to solve this question about binomial theorem, could someone help me? :)

Consider the expansion of \(\displaystyle x^2\left(3x^2+\frac{k}{x}\right)^8\)
The constant term is 16128.
Find k.
 
Last edited:
Mathematics news on Phys.org
  • #2
Hello and welcome to MHB, pusekatja! :D

Without knowing what you have tried, I can't know where you are stuck, so this is why we ask our users to show what they have tried so far.

Now, according to the binomial theorem, the $i$th term of the given expansion will be:

\(\displaystyle x^2{8 \choose i}\left(3x^2\right)^{8-i}\left(\frac{k}{x}\right)^i\)

How about we rewrite this term as follows:

\(\displaystyle 3^{8-i}{8 \choose i}k^ix^{18-3i}\)

Now, what does this tell us $i$ must be in order for the term to be a constant?
 

FAQ: Help to solve binominal theorem

What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand binomials, which are expressions with two terms, raised to a given power. It is also known as the binomial expansion or binomial formula.

Why is the binomial theorem important?

The binomial theorem is important because it allows us to easily expand and simplify binomial expressions, which are frequently used in many areas of mathematics and science. It also helps in solving problems involving probability and statistics.

How do you use the binomial theorem?

To use the binomial theorem, you need to know the formula, which states that (a + b)^n = ∑(n choose k) * a^(n-k) * b^k, where n is the power, a and b are the two terms, and k is the index of summation. You can then plug in the values and simplify the expression using combinatorial techniques.

What are some real-life applications of the binomial theorem?

The binomial theorem has various applications in fields such as physics, chemistry, engineering, and economics. For example, it can be used to calculate the probabilities of different outcomes in a series of events, determine the expansion of a gas under changing conditions, or approximate the future value of an investment.

Are there any limitations to the binomial theorem?

Yes, the binomial theorem has some limitations. It can only be applied to binomials, which means expressions with two terms. It also assumes that the terms are raised to a whole number power. Furthermore, it may not be accurate for large values of the power or non-integer powers.

Similar threads

MHB Solve Binominal Form (4x+3)^n | Binomial Coefficients
Replies
2
Views
1K
B Intermediate Value Theorem and Synthetic Division
Replies
4
Views
2K
MHB Linear combination of sine and cosine function
Replies
3
Views
1K
MHB Fixed point iteration convergence
Replies
1
Views
1K
Binomial Theorem - determine the term with...
Replies
18
Views
2K
How can I rederive this Mathematica result for a complex equation?
Replies
2
Views
1K
A Help needed with derivation: solving a complex double integral
Replies
1
Views
1K
MHB Can the Partial Sum of a Difficult Series be Solved with Induction?
Replies
6
Views
2K
MHB Vandomo's question at Yahoo Answers regarding the binomial theorem
Replies
1
Views
2K
I Trig Manipulations I'm Not Getting
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top