The expression `(sqrt(2x^2+1)+sqrt(2x^2-1))^6 + (2/(sqrt(2x^2+1)+sqrt(2x^2-1)))^6` is polynomial of degree

Options

A. 6
B. 8
C. 10
D. 12

Answer

A

Explanation

We have,
`(2)/(sqrt(2x^(2)+1)+sqrt(2x^(2)-1)) = (2(sqrt(2x^(2)+1)-sqrt(2x^(2)-1)))/((2x^(2)+1)-(2x^(2)-1))`
`= sqrt(2x^(2)+1)-sqrt(2x^(2)-1)`
Thus, the given expression cen be written as
`(sqrt(2x^(2)+1)+sqrt(2x^(2)-1))^(6)+(sqrt(2x^(2)+1)-sqrt(2x^(2)-1))^(6)`
But `(a+b)^(6)(a-b)^(6) = 2[a^(6)+.^(6)C_(2)a^(4)b^(4)b^(2)+.^(6)C_(4)a^(2)b^(4)+b^(6)]`
Therefore,
`(sqrt(2x^(2)+1)+sqrt(2x^(2)-1))^(6) + (sqrt(2x^(2)+1)-sqrt(2x^(2)-1))^(6)`
`= 2[(2x^(2)+1)^(3)+15(2x^(2)+1)^(2)(2x^(2)-1) + 15(2x^(3)+1)xx(2x^(2)-1)^(2)+(2x^(2)--1)^(3)]`
Which is polynomial of degree 6.