For a real polynomial in one variable $P$, let $Z ( P )$ denote the locus of points ( $x , y$ ) in the plane such that $P ( x ) + P ( y ) = 0$. Then, (A) there exist polynomials $Q _ { 1 }$ and $Q _ { 2 }$ such that $Z \left( Q _ { 1 } \right)$ is a circle and $Z \left( Q _ { 2 } \right)$ is a parabola. (B) there does not exist any polynomial $Q$ such that $Z ( Q )$ is a circle or a parabola. (C) there exists a polynomial $Q$ such that $Z ( Q )$ is a circle but there does not exist any polynomial $P$ such that $Z ( P )$ is a parabola. (D) there exists a polynomial $Q$ such that $Z ( Q )$ is a parabola but there does not exist any polynomial $P$ such that $Z ( P )$ is a circle.
For a real polynomial in one variable $P$, let $Z ( P )$ denote the locus of points ( $x , y$ ) in the plane such that $P ( x ) + P ( y ) = 0$. Then,\\
(A) there exist polynomials $Q _ { 1 }$ and $Q _ { 2 }$ such that $Z \left( Q _ { 1 } \right)$ is a circle and $Z \left( Q _ { 2 } \right)$ is a parabola.\\
(B) there does not exist any polynomial $Q$ such that $Z ( Q )$ is a circle or a parabola.\\
(C) there exists a polynomial $Q$ such that $Z ( Q )$ is a circle but there does not exist any polynomial $P$ such that $Z ( P )$ is a parabola.\\
(D) there exists a polynomial $Q$ such that $Z ( Q )$ is a parabola but there does not exist any polynomial $P$ such that $Z ( P )$ is a circle.