[12 points] Consider polynomials $p(x)$ with the following property, called $(\dagger)$. $(\dagger)$ If $r$ is a root of $p(x)$, then $r^{2} - 4$ is also a root of $p(x)$. (i) We want to find every quadratic polynomial of the form $p(x) = x^{2} + bx + c$ such that $p(x)$ has two distinct roots, has integer coefficients and has property $(\dagger)$. Prove that there are exactly two such polynomials and list them. (ii) It is also true that there are exactly two cubic polynomials of the form $p(x) = x^{3} + ax^{2} + bx + c$ with the property $(\dagger)$ such that $p(x)$ shares no root with the polynomials you found in part (i). Explain fully how you will prove this along with the method to find the polynomials, but do not try to explicitly find the polynomials.
[12 points] Consider polynomials $p(x)$ with the following property, called $(\dagger)$.\\
$(\dagger)$ If $r$ is a root of $p(x)$, then $r^{2} - 4$ is also a root of $p(x)$.\\
(i) We want to find every quadratic polynomial of the form $p(x) = x^{2} + bx + c$ such that $p(x)$ has two distinct roots, has integer coefficients and has property $(\dagger)$. Prove that there are exactly two such polynomials and list them.\\
(ii) It is also true that there are exactly two cubic polynomials of the form $p(x) = x^{3} + ax^{2} + bx + c$ with the property $(\dagger)$ such that $p(x)$ shares no root with the polynomials you found in part (i). Explain fully how you will prove this along with the method to find the polynomials, but do not try to explicitly find the polynomials.