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Let y = ( x 3 − 1 ) 2 1 . Then:
\left(x + (x^3-1)^{\frac{1}{2}} \right)^5 + \left(x - (x^3-1)^{\frac{1}{2}} \right)^5 \\ = \left(x + y \right)^5 + \left(x - y \right)^5 \\ = x^5 \color{#D61F06}{+ 5x^4y} + 10 x^3y^2 \color{#D61F06}{+ 10 x^2y^3} + 5xy^4 \color{#D61F06}{+ y^5} + x^5 \color{#D61F06}{- 5x^4y} + 10 x^3y^2 \color{#D61F06}{- 10 x^2y^3} + 5xy^4 \color{#D61F06}{- y^5} \\ = 2x^5 + 20 x^3y^2 + 10xy^4 \\ = 2x^5 + 20 x^3(x^3-1) + 10\color{#3D99F6}{x(x^3-1)^2} \\ = 2x^5 + 20 x^3(x^3-1) + 10(\color{#3D99F6}{x^\boxed{7}-2x^4+x})
Therefore, the degree of the polynomial is 7 .