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Q). The expansion ${\left\{x+{\left(x^3-1\right)}^{1/2}\right\}}^5+{\left\{x-{\left(x^3-1\right)}^{1/2}\right\}}^5$ is a polynomial of degree
(a) 5
(b) 6
(c) 7
(d) 8

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Let} {\left({\mathrm{x}}^3-1\right)}^{\frac{1}{2}}=\mathrm{y} \\ \mathrm{Hence} \\ {\left(\mathrm{x}+\mathrm{y}\right)}^5+{\left(\mathrm{x}-\mathrm{y}\right)}^5 \\ =\left({}^5{\mathrm{C}}_0{\mathrm{x}}^5{+}^5{\mathrm{C}}_1{\mathrm{x}}^4\mathrm{y}{+}^5{\mathrm{C}}_2{\mathrm{x}}^3{\mathrm{y}}^2{+}^5{\mathrm{C}}_3{\mathrm{x}}^2{\mathrm{y}}^3{+}^5{\mathrm{C}}_4{\mathrm{x}}^1{\mathrm{y}}^4{+}^5{\mathrm{C}}_5{\mathrm{y}}^5\right)+\left({}^5{\mathrm{C}}_0{\mathrm{x}}^5{-}^5{\mathrm{C}}_1{\mathrm{x}}^4\mathrm{y}{+}^5{\mathrm{C}}_2{\mathrm{x}}^3{\mathrm{y}}^2{-}^5{\mathrm{C}}_3{\mathrm{x}}^2{\mathrm{y}}^3{+}^5{\mathrm{C}}_4{\mathrm{x}}^1{\mathrm{y}}^4{-}^5{\mathrm{C}}_5{\mathrm{y}}^5\right) \\ =2\left({}^5{\mathrm{C}}_0{\mathrm{x}}^5{+}^5{\mathrm{C}}_2{\mathrm{x}}^3{\mathrm{y}}^2{+}^5{\mathrm{C}}_4{\mathrm{x}}^1{\mathrm{y}}^4\right) \\ =2\left({}^5{\mathrm{C}}_0{\mathrm{x}}^5{+}^5{\mathrm{C}}_2{\mathrm{x}}^3\left({\mathrm{x}}^3-1\right){+}^5{\mathrm{C}}_4{\mathrm{x}}^1{\left({\mathrm{x}}^3-1\right)}^2\right) \\ \mathrm{Now} \mathrm{clear} \mathrm{degree} \mathrm{of} \mathrm{last} \mathrm{term} \mathrm{is} \mathrm{highest} {\to }^5{\mathrm{C}}_4{\mathrm{x}}^1{\left({\mathrm{x}}^3-1\right)}^2 \mathrm{which} \mathrm{is} 1+2\times 3=7 \end{gathered}$$

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