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Term Independent of x

Find the term...

Question

Find the term independent of $x$ in $(2 + 3 x - 4 x^{2}) {(x + \frac{1}{x})}^{10}$

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Solution

Given, ${(x + \frac{1}{x})}^{10}$

We know that general term is $t_{r + 1} =^{n} C_{r} X^{n - r} . a^{r}$

Where n= 10 , X=x, a= $\frac{1}{x}$

Hence,

$t_{r + 1} =^{10} C_{r} x^{10 - r} . {(\frac{1}{x})}^{r}$

$t_{r + 1} =^{10} C_{r} x^{10 - r} . \frac{1}{x^{r}}$

$=^{10} C_{r} {x^{10 - r}}^{- r}$

$=^{10} C_{r} x^{10 - 2 r}$

For independent term of x

$x^{10 - 2 r} = x^{0}$

By compare of power of x we get

10-2r=0

r=5

Hence,

$t_{5 + 1} =^{10} C_{5} x^{10 - 2 \times r}$

$t_{6} =^{10} C_{5} x^{0}$

$t_{6} =^{10} C_{5}$

6^th term is independent of x , which is $^{10} C_{6}$

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