Homework Help: Determine the term containing (Binomial Theorom)

1. Dec 17, 2012

Determine the term containing.... (Binomial Theorom)

1. The problem statement, all variables and given/known data

Determine the term containing x^11 in the expansion (X^2 - 1/x)^10

2. Relevant equations

T(k+1) = nC(k)*A^(n-k)*b^K

3. The attempt at a solution

x^11 = 10Ck * (x^2)^(10-k)*(-X^-1)^k (taking out the (-) from the x^-1 b/c it dont matter)

x^11 = 10ck *X^(20-2k)(X^-k)
x^11/(X^20-3k) = 10ck
x^3k-9 = 10ck

Im stuck here.

Last edited: Dec 17, 2012
2. Dec 17, 2012

haruspex

Re: Determine the term containing.... (Binomial Theorom)

Start by finding the coefficient of x1. That should give you a hint.

3. Dec 17, 2012

Re: Determine the term containing.... (Binomial Theorom)

Its just 1. How does that help me ?

4. Dec 17, 2012

Dick

Re: Determine the term containing.... (Binomial Theorom)

I don't think it is 1. $(x^2-\frac{1}{10})^{10}$ contains only even powers of x, doesn't it? Forget the binomial theorem and just imagine what would happen if you start expanding it by hand.

5. Dec 17, 2012

Re: Determine the term containing.... (Binomial Theorom)

Mistake in my question.

6. Dec 17, 2012

Dick

Re: Determine the term containing.... (Binomial Theorom)

Yeah. So you want $(x^2-\frac{1}{x})^{10}$. Factor out the x^2. So now you've got $x^{20} (1-\frac{1}{x^3})^{10}$. What power do you want in the second factor?

7. Dec 17, 2012

Re: Determine the term containing.... (Binomial Theorom)

Woah how did you get that? And im looking for the term with X^11.

8. Dec 18, 2012

Dick

Re: Determine the term containing.... (Binomial Theorom)

I used algebra. And to go back to your first post, which was pretty hard to read especially with the wrong problem statement, you want 20-3k=11. What's k? And the sign does matter.

9. Dec 18, 2012

Re: Determine the term containing.... (Binomial Theorom)

Ok, so I'm looking for the term containing X^11. I tried using the formula T(k+1) = nC(k)*A^(n-k)*b^K. K is the variable of the formula.

10. Dec 18, 2012

Re: Determine the term containing.... (Binomial Theorom)

n is 10.

11. Dec 18, 2012

Re: Determine the term containing.... (Binomial Theorom)

I know the answer is 4 b/c it would be. 10c3(x^2)^7(-1/x)^3. However I dont care what the constant term is.

12. Dec 18, 2012

Re: Determine the term containing.... (Binomial Theorom)

K is 3 making the term 4 according to the formula T(k+1) = nC(k)*A^(n-k)*b^K.

13. Dec 18, 2012

Dick

Re: Determine the term containing.... (Binomial Theorom)

You were almost there is the first post. You want the exponent in (x^2)^(10-k)*(x^(-k)) to be 11. What's k?

14. Dec 18, 2012

Dick

Re: Determine the term containing.... (Binomial Theorom)

Yes, k=3. Now think a little about the sign of the coefficient.

15. Dec 18, 2012

Re: Determine the term containing.... (Binomial Theorom)

Yes its negative, however cant I just dismiss the coefficient since i'm only looking for its variable ?

16. Dec 18, 2012

haruspex

Re: Determine the term containing.... (Binomial Theorom)

Dick took a factor of x2 out of each of the 10 brackets. That turned each bracket into 1-1/x3, and combining the 10 factors outside produced x20.
Now imagine expanding the (1-1/x3)10. Suppose some term of that has the x power as 1/xk. You want the one which, when multiplied by x20, produces x11.

17. Dec 18, 2012

Dick

Re: Determine the term containing.... (Binomial Theorom)

You threw away the sign and got 10C3 once you figured out k=3. But you shouldn't have thrown away the sign. It might be -10C3.

18. Dec 18, 2012

Re: Determine the term containing.... (Binomial Theorom)

I like how this was solved with mental thinking. But how would you solve it using sheer algebra ?

19. Dec 18, 2012

Re: Determine the term containing.... (Binomial Theorom)

You know, what you guys showed / helped me with is enough to solve similar questions. Thank you. I didnt know you can take x^2 out of (x^2 - 1/x)^10

20. Dec 18, 2012

Dick

Re: Determine the term containing.... (Binomial Theorom)

You very nearly already did it in the first post. Add the insight that k must be 3, don't ignore the sign and try to write it out in a coherent way.
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