Bring radical

Remarks

The definition makes sense: $g(x) = x^5 + x + a$ is strictly increasing on $\R$, since $g'(x) = 5x^4 + 1 > 0$, and it tends to $\pm\infty$ as $x \to \pm\infty$, so it has exactly one real root. Example: $\operatorname{BR}(-2) = 1$, since $1^5 + 1 - 2 = 0$; and $\operatorname{BR}(0) = 0$. Non-example: for the polynomial $x^5 - x + a$, the derivative $5x^4 - 1$ changes sign, and for $a = 0$ there are three real roots $-1, 0, 1$, so no "unique real root" is defined in the same way. The function $\operatorname{BR}$ is odd, continuous and strictly decreasing on $\R$.