If and be the coefficients of x^4 and x^2 respectively in the expansion of (x+x^2-1)^6+(x-…

Question

If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of $\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6$, then:

Accepted Answer

$$\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6 = 2\left[\binom60x^6+\binom62x^4(x^2-1)+\binom64x^2(x^2-1)^2+\binom66(x^2-1)^3\right]$$

$$=2\left[x^6+15(x^6-x^4)+15x^2(x^4-2x^2+1)+(x^6-3x^4+3x^2-1)\right]$$

$$=2\left(32x^6-48x^4+18x^2-1\right)$$

So $\alpha$ (coeff of $x^4$) $=-96$, $\beta$ (coeff of $x^2$) $=36$.

$$\alpha-\beta=-96-36=-132$$

Solutions · 1