# Binomial Expansion

Find the coefficient of the term X^5 of the expansion
$$(3x^3 - \frac{1}{x^2})^{10}$$

Another question off the topic.
Find the x-coordinate of the minimium point of $$y=2x^2-5x+3$$
I know I have to complete the square but I'm not sure how its done.

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## Answers and Replies

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?

I know calculus. So how do I do it using calculus?

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

Nope. Never heard of that.

dextercioby
Science Advisor
Homework Helper
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
$$C_{n}^{k}a^{k}b^{n-k}$$

Daniel.

But I forgot how to complete the square.

dextercioby
Science Advisor
Homework Helper
$$ax^{2}+bx+c=a(x^{2}+\frac{b}{a}x)+c=a[x^{2}+2\cdot (\frac{b}{2a})\cdot x+(\frac{b}{2a})^{2}]+c-a(\frac{b}{2a})^{2}=a(x+\frac{b}{2a})^{2}+c-\frac{b^{2}}{4a}$$

Apply it.

Daniel.

Thanks for the help!

dextercioby
Science Advisor
Homework Helper
You're welcome.I hope you will master "completing the square",eventually...

Daniel.

P.S.It would be embarrasing to use calculus to find the maximum/minimum of a parabola...

dextercioby said:
P.S.It would be embarrasing to use calculus to find the maximum/minimum of a parabola...

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:.