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Find the term independent of x in (x+1x)4.

Answer
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Hint: First of all by using the formula for general terms of binomial expansion that is, Tn=nCranrbr, write the general term of (x+1x)4by taking a=x,b=1x and n=4. Then put the power of x = 0 to find the term independent of x.

Complete step-by-step answer:
Let us consider the expression given in question as
A=(x+1x)4…………… (1)

We know that, by binomial theorem, we can expand (a+b)n as,
(a+b)n=nC1an1b1+nC2an2b2+nC3an3b3+.......nCna1bn1

We can also write it as,
(a+b)n=r=1nnCranrbr

Therefore, we get general term in expansion of
(a+b)n as nCranrbr

By taking a=x,b=1x&n=4we get general term of (x+1x)4as,
General term of (x+1x)4=4Cr(x)4r(1x)r

Let us consider this general term as,
Tn=4Cr(x)4r[1x]r

We know that (ab)n=anbn , applying this in above expression, we get,
Tn=4Cr(x)4r(1)r(x)r
Or we get Tn=4Cr(x)4r1(x)r

We can also write above expression as,
Tn=4Cr(x)4r(x)r

Now, we know that apaq=apq. By applying this in above expression, we get
 Tn=4Cr(x)(4r)r
Therefore, we get Tn=4Cr(x)42r……………. (2)

Now, to find the term which is independent of x, we must put the power of x = 0.
Therefore we get, 4-2r=0
By taking the terms containing ‘r’ to one side and constant term to other side, we get,
2r=4

By dividing 2 on both sides, we get,
2r2=42r=2
Now to get the term independent of x, we put r= 2 in equation (2), we get,
Tn=4C2(x)42(2)Tn=4C2x44Tn=4C2(x)0
We know that (a)0=1, therefore we get.
Tn=4C2
Tn=4!2!2!Tn=6
Therefore, we get the term independent of x in (x+1x)4as 4C2=6.

Note: Students must note that when they are asked to find the term independent of variable, they should always put the power of that variable = 0. Also students must take special care while writing each term and cross check if they have written it correctly or not. Students often make mistakes while writing the powers and this must be avoided.