# Binomial Expansion

1. Jan 22, 2005

### footprints

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.

Last edited: Jan 22, 2005
2. Jan 22, 2005

### maverick280857

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. Jan 22, 2005

### footprints

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

4. Jan 22, 2005

### maverick280857

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.

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. Jan 22, 2005

### footprints

Nope. Never heard of that.

6. Jan 22, 2005

### dextercioby

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.

7. Jan 22, 2005

### footprints

But I forgot how to complete the square.

8. Jan 22, 2005

### dextercioby

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

Apply it.

Daniel.

9. Jan 22, 2005

### footprints

Thanks for the help!

10. Jan 22, 2005

### dextercioby

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...

11. Jan 27, 2005

### maverick280857

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:.