Consider polynomials of the form $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$ where $a , b , c$ are integers. Name the three (possibly non-real) roots of $f ( x )$ to be $p , q , r$. (a) If $f ( 1 ) = 2021$, then $f ( x ) = ( x - 1 ) \left( x ^ { 2 } + s x + t \right) + 2021$ where $s , t$ must be integers. (b) There is such a polynomial $f ( x )$ with $c = 2021$ and $p = 2$. (c) There is such a polynomial $f ( x )$ with $r = \frac { 1 } { 2 }$. (d) The value of $p ^ { 2 } + q ^ { 2 } + r ^ { 2 }$ does not depend on the value of $c$.
Consider polynomials of the form $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$ where $a , b , c$ are integers. Name the three (possibly non-real) roots of $f ( x )$ to be $p , q , r$.
(a) If $f ( 1 ) = 2021$, then $f ( x ) = ( x - 1 ) \left( x ^ { 2 } + s x + t \right) + 2021$ where $s , t$ must be integers.\\
(b) There is such a polynomial $f ( x )$ with $c = 2021$ and $p = 2$.\\
(c) There is such a polynomial $f ( x )$ with $r = \frac { 1 } { 2 }$.\\
(d) The value of $p ^ { 2 } + q ^ { 2 } + r ^ { 2 }$ does not depend on the value of $c$.