Question

Find the constant term in the expansion of (x +1/x)'8​

Answer 1

`\left( x+(1)/(x)\right)^(8) =\sum _(k=0)^(8)\binom{8}{k} \cdotp x^(8-k) \cdotp \left((1)/(x)\right)^(k)\\\\\\ \begin{array}{l}\text{We\ wish\ to\ find\ } x^(0) :\\\\x^(8-k) \cdotp \left((1)/(x)\right)^(k) =x^(0)\\\\\Longrightarrow x^(8-k) \cdotp x^(-k) =x^(0)\\\\\Longrightarrow \cancel{x}^(8-2k) =\cancel{x}^(0)\\\\\Longrightarrow 2k=8\\\\\Longrightarrow k=4\end{array}\\\\\therefore \binom{8}{4} \cdotp x^(4) \cdotp \left((1)/(x)\right)^(4) =(8!)/(( 8-4) !\cdotp 4!)`

`\binom{8}{4} \cdotp x^(4) \cdotp \left((1)/(x)\right)^(4) =\frac{8\cdotp 7\cdotp 6\cdotp 5\cdotp \cancel{4!}}{\cancel{4!} \cdotp 4\cdotp 3\cdotp 2\cdotp 1}\\\\\binom{8}{4} \cdotp x^(4) \cdotp \left((1)/(x)\right)^(4)=70\\\\\therefore \boxed{\boxed{\text{The\ constant\ term\ is\ } 70}}`