Solve Binomial Theorem: Find Term Independent of x

AI Thread Summary
The discussion focuses on finding the term independent of x in the expansion of (2+x)[2x+(1/x)]^5 using the Binomial theorem. Participants clarify that a term independent of x can arise from multiplying a constant term with another constant or from a term with x^-1 multiplied by a term with x. The key realization is that the constant term must be derived from the expansion of (2+x) and the relevant term from (2x+(1/x))^5. The calculations reveal that the term independent of x occurs when r=3, leading to the conclusion that the third term in the expansion is the desired result. Understanding this concept is essential for solving similar problems involving the Binomial theorem.
thornluke
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Homework Statement
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.
 
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thornluke said:
Homework Statement
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.

What would be happen if it becomes zero? See there's a term (2+x) also present. :wink:
 
Pranav-Arora said:
What would be happen if it becomes zero? See there's a term (2+x) also present. :wink:

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

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

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.
 
Infinitum said:
Because the term already has a x-1! It cannot be independent of x if it already has a x :rolleyes:

I am so lost here.. why am I so terrible at maths :frown:
 
thornluke said:
Constant term means the term with no x right?
Yes! :smile:

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..
 
Pranav-Arora said:
Yes! :smile:

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..

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?
 
thornluke said:
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?

What is (2+(-0.1))5=?
 
Pranav-Arora said:
What is (2+(-0.1))5=?

Thanks very much. Can't believe I can't even think simple.
 
  • #10
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.
 
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