Question

Let $f\left(x\right)=(\sqrt{x^2+1}+\sqrt{x^2-1}{)}^6+{\left(\frac{2}{\sqrt{x^2+1}+\sqrt{x^2-1}}\right)}^6$.Then

• A. $f\left(x\right)$ is a polynomial of sixth degree in $x$
• B. $f\left(x\right)$ has exactly two terms
• C. $f\left(x\right)$ is not a polynomial in $x$
• D. coefficient of $x^6$ is $64$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $f\left(x\right)$ is a polynomial of sixth degree in $x$
B $f\left(x\right)$ has exactly two terms
D coefficient of $x^6$ is $64$

Let $x^2+1=a,x^2-1=b$
Therefore the given expression can be written as,

$\frac{2}{\sqrt{a}+\sqrt{b}}=\sqrt{a}-\sqrt{b}$
${(\sqrt{a}+\sqrt{b})}^6+{(\sqrt{a}-\sqrt{b})}^6=2\times (a^3+b^3+15\times a\times b\times (a+b\left)\right)$
so,required solution is $64\times x^6-48\times x^2$
so,options a,b,d are correct