Quantcast

A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Grazes

  • 2 years ago

Find the term in the binomial expansion of (((x^2)/3)-(1/x))^9 that is a constant.

  • This Question is Closed
  1. Grazes
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[(\frac{ x ^{2} }{ 3 } - \frac{ 1 }{ x })^{9}\]

  2. hewsmike
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Nasty one. The binomial expansion\[(a+b)^{n}= \sum_{k=0}^{n}\frac{ n! }{k!(n-k)!}a^{n-k}b^{k}\]has powers of a decreasing while those of b increase as you go up in the index k. Consider for this problem that \[a=x^{2}/3\]and \[b= -1/x=-x^{-1}\]and \[n=9\]They are effectively asking for the value of k that gives\[(a)^{n-k}(b)^{k}=(x^{2}/3)^{9-k}(-x^{-1})^{k}=constant \times x^{0}\]and so looking at those powers\[2(9-k)+(-k)=0\]\[18-2k-k=0\]\[3k=18\]\[k=3\]so what is the multiplier of that power of x?\[(1/3)^{9-3}(-1)^{3}\frac{9!}{3!(9-3)!}=(-1)(3^{-6})\frac{(9\times 8 \times 7 \times 6 \times 5\times4 \times 3 \times 2 \times 1)}{(3 \times 2 \times 1)(6 \times 5\times4 \times 3 \times 2 \times 1)}\].... aaargh ......\[=-28/3^{5}\]I think ! :-)

  3. Grazes
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I just know that the final answer is 29/8. Thanks!

  4. hewsmike
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Ah, I've probably fumbled the arithmetic somewhere, but the method is as described. :-)

  5. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.