Binomial Expansion

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  • #1
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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.
 
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  • #2
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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?
 
  • #3
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I know calculus. So how do I do it using calculus?
 
  • #4
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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
 
  • #5
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Nope. Never heard of that.
 
  • #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.
 
  • #7
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But I forgot how to complete the square. :redface:
 
  • #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.
 
  • #9
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Thanks for the help!
 
  • #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:
 
  • #11
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dextercioby said:
P.S.It would be embarrasing to use calculus to find the maximum/minimum of a parabola... :rolleyes:
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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