The expansion of ${[x+{(x^3-1)}^{\frac{1}{2}}]}^5+{[x-{(x^3-1)}^{\frac{1}{2}}]}^5$ is a polynomial of degree 

${\left[x+{(x^3-1)}^{\frac{1}{2}}\right]}^5+{\left[x-{(x^3-1)}^{\frac{{\displaystyle 1}}{{\displaystyle 2}}}\right]}^5$

$=2\left[{ }^5{\text{C}}_0{x}^5+{}^5{\text{C}}_2{x}^3(x^3-1)+{}^5{\text{C}}_{4}x{(x^3-1)}^2\right]$

$=2[x^5+10x^6-10x^3+5x(x^6-2x^3+1)]$

$=2[x^5-10x^3+10x^6+5x^7-10x^4+5x]$

$=10x^7+20x^6+2x^5-20x^4-20x^3+10x$
$\therefore$ Polynomial degree 7