Question

The given expression is ${(x+\sqrt{x^3-1})}^5+{(x-\sqrt{x^3-1})}^5$ is a

• A. polynomial of fractional degree.
• B. polynomial of degree 7.
• C. polynomial with integer degree.
• D. not a polynomial.

Answer 1

Answer: (B), (C)

The correct options are
B polynomial of degree 7.
C polynomial with integer degree.

The given expression is ${(x+\sqrt{x^3-1})}^5+{(x-\sqrt{x^3-1})}^5$ we know that
$(x+a{)}^n+(x-a{)}^n=2{[}^n{C}_0{x}^n{+}^n{C}_2{x}^{n-2}{a}^2{+}^n{C}^4{x}^{n-4}{a}^4+\cdot ]$
Therefore the given expression is equal to $2{[}^5{C}_0{x}^5{+}^5{C}_2{x}^3(x^3-1){+}^5{C}_{4}x(x^3-1{)}^2].$
Maximum power of $x$ involved here is 7.
Also only + ve integral powers of $x$ are involved.
Therefore, the given expression is a polynomial of degree 7.