For |x| < 1 2 , if the coefficient of x¹⁰ and the constant term in the expansion of 2 x^3+8 x^2-2 x-2 (1-x)(1+x)(1-2 x) in powers of x are l and m respectively, then 1 m=

Step-by-step solution

The given expression is 2 x^3+8 x^2-2 x-2 (1-x)(1+x)(1-2 x) = 2 x^3+8 x^2-2 x-2 (2 x-1)(x-1)(x+1) =1+ A 2 x-1 + B x-1 + C x+1 (By partial fraction) aligned 2 x^3+8 x^2-2 x & -2 = (2 x^3 . & .-x^2-2 x+1 )+A (x^2-1 ) & +B (2 x^2+x-1 )+C (2 x^2-3 x+1 ) aligned On comparing, we get aligned -1+A+2 B+2 C & =8 -2+B-3 C & =-2 aligned and 1-A-B+C=-2 On solving we have A=1, B=3 and C=1 Given expression is aligned & 2 x^3+8 x^2-2 x-2 (1-x)(1+x)(1-2 x) =1- 1 1-2 x - 3 1-x + 1 1+x = & 1-(1-2 x)⁻¹-3(1-x)⁻¹+(1+x)⁻¹ aligned Coeffi