# Binomial Theorem

1. Jun 2, 2012

### thornluke

The problem statement, all variables and given/known data
The method of Binomial expansion is useful because you can avoid expanding large expressions:
Q: Find the term indepedent of x in the expansion of (2+x)[2x+(1/x)]5

The attempt at a solution:
"For this to produce a term independent of x, the expansion of [2x+(1/x)]5 must have a constant term or a term in x-1...?
So the power of x is given by 5-2r. This cannot be zero for positive integer values of r. Hence the required coefficient is given by:
5 - 2r = -1
r = 3"

But why?! I do not understand that underlined statement.

2. Jun 2, 2012

### Saitama

What would be happen if it becomes zero? See there's a term (2+x) also present.

3. Jun 2, 2012

### thornluke

So...the expansion of [2x+(1/x)]5 must have a constant term or a term in x-1...

Constant term means the term with no x right?

But why can the term independent of x be in x-1 too?

4. Jun 2, 2012

### Infinitum

Because the term already has a x-1!! It cannot be independent of x if it already has a x

To explain the question, you need a term independent of x. You get them independent of x, if a constant term gets multiplied with another constant term, OR a term with x-1 is multiplied with a term with x. But there is no constant term(not involving x) in the expansion of [2x+(1/x)]5 as explained. So the only possibility remains, that x-1 gets multiplied with x.

5. Jun 2, 2012

### thornluke

I am so lost here.. why am I so terrible at maths

6. Jun 2, 2012

### Saitama

Yes!

We need to find the constant term in expansion of (2+x)(2x+(1/x))5, not in (2x+(1/x))5. So when x in (2+x) multiplies by a term having x-1 with it, we get a constant term as $x*\frac{1}{x}=1.$.

7. Jun 2, 2012

### thornluke

Ahh.. I think I'm starting get the grasp of this.. one more problem!
Expand (2+x)5 and hence find 1.95
= 32 + 80x+ 80x2 + 40x3 + 10x4 + x5

Why can 1.95 be considered to be x= -0.1 in the above expansion?

8. Jun 2, 2012

### Saitama

What is (2+(-0.1))5=?

9. Jun 2, 2012

### thornluke

Thanks very much. Can't believe I can't even think simple.

10. Jun 5, 2012

### cool_jessica

its binomial theorem.
use the formula t(r+1)=nCr*x^n-r*a6r
x=x^2 and a=-1/2x (substitution of the question variables)
n=power=3
let the term independent of x be t(r+1)=x^0=1
therefore after substitution the formula would be
Y*x^0=nCr*(x^2)^3-r*(-1*x^ -1)^r
therefore power of x=6-2r-r
but in the term power of x is supposed to be 0
0=6-3r
6=3r
r=2.
therefore the term is r+1
2+1
=3..
therefore the term independent of x is the third term.