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Binomial Expansion

  1. Jan 22, 2005 #1
    Find the coefficient of the term X^5 of the expansion
    [tex](3x^3 - \frac{1}{x^2})^{10}[/tex]

    Another question off the topic.
    Find the x-coordinate of the minimium point of [tex]y=2x^2-5x+3[/tex]
    I know I have to complete the square but I'm not sure how its done.
     
    Last edited: Jan 22, 2005
  2. jcsd
  3. Jan 22, 2005 #2
    Can you write the general term in the expansion of (x+y)^n? If you can, then you can replace x by 3x^3 and y by (-1/x^2).

    For the second question, completing the square is a good idea if you do not know calculus (or are not supposed to use it).

    Why not show your solution first?
     
  4. Jan 22, 2005 #3
    I know calculus. So how do I do it using calculus?
     
  5. Jan 22, 2005 #4
    Do you know the First and Second Derivative Tests?

    What happens to a continuous function when its derivative switches sign? By a theorem called the Intermediate Value Theorem, every function which switches sign at least once over an interval must attain the value zero.

    Try sketching a graph to convince yourself about the behavior of your quadratic polynomial.

    At this point, you should consult your Calculus textbook for the First and Second Derivative Tests. If you have a problem, I'd be glad to help further.

    Cheers
    Vivek
     
  6. Jan 22, 2005 #5
    Nope. Never heard of that.
     
  7. Jan 22, 2005 #6

    dextercioby

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    Okay,forget about calculus.This is elementary.Take the previous advice to complete the square.

    As for the first problem:The general term in the binomial expasion is
    [tex] C_{n}^{k}a^{k}b^{n-k} [/tex]

    Daniel.
     
  8. Jan 22, 2005 #7
    But I forgot how to complete the square. :redface:
     
  9. Jan 22, 2005 #8

    dextercioby

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    [tex] ax^{2}+bx+c=a(x^{2}+\frac{b}{a}x)+c=a[x^{2}+2\cdot (\frac{b}{2a})\cdot x+(\frac{b}{2a})^{2}]+c-a(\frac{b}{2a})^{2}=a(x+\frac{b}{2a})^{2}+c-\frac{b^{2}}{4a} [/tex]

    Apply it.

    Daniel.
     
  10. Jan 22, 2005 #9
    Thanks for the help!
     
  11. Jan 22, 2005 #10

    dextercioby

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    You're welcome.I hope you will master "completing the square",eventually... :smile:

    Daniel.

    P.S.It would be embarrasing to use calculus to find the maximum/minimum of a parabola... :rolleyes:
     
  12. Jan 27, 2005 #11
    I couldn't agree more...but you know its way faster than completing the square (you can write the answer by inspection and this is an asset if you're in a hurry). Nevertheless, its embarrasing :tongue:.
     
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