Find-the-coefficient-of-the-term-independent-of-x-in-the-expansion-of-x-1-x-2-3-x-1-3-1-x-1-x-x-1-2-10-

Question Number 9747 by tawakalitu last updated on 30/Dec/16

Find the coefficient of the term independent

of x in the expansion of

{(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}

Answered by sandy_suhendra last updated on 30/Dec/16

using the formula :

a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})

a^{2} - b^{2} = (a + b) (a - b)

{[\frac{{(x^{1 / 3})}^{3} + 1^{3}}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{{(x^{1 / 2})}^{2} - 1^{2}}{x^{1 / 2} (x^{1 / 2} - 1)}]}^{10}

= {[\frac{(x^{1 / 3} + 1) (x^{2 / 3} - x^{1 / 3} + 1)}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{(x^{1 / 2} + 1) (x^{1 / 2} - 1)}{x^{1 / 2} (x^{1 / 2} - 1)}]}^{10}

= {[x^{1 / 3} + 1 - 1 - \frac{1}{x^{1 / 2}}]}^{10}

= {[x^{1 / 3} - \frac{1}{x^{1 / 2}}]}^{10}

the term independent of x = x^{0}

\Rightarrow 10 C 6 {(x^{1 / 3})}^{6} {(- \frac{1}{x^{1 / 2}})}^{4} = 210 x^{0}

so the coefficient is 210

Commented by tawakalitu last updated on 30/Dec/16

thank you sir . God bless you .