Solutions

a) By Problem 3-7 d), there exists a polynomial function $f$ of degree $n-1$ with $n-1$ roots. Hence, $f(x)=0$ for precisely $n-1$ numbers $x$. Moreover, by Problem 10-22 a), there exists a polynomial function $g$ of degree $n$ such that $g'=f$.

b) There exists a polynomial function $f$ of degree $n-1$, with $n-1$ being even, and having no roots. Hence, $f(x)=0$ for no $x$. Moreover, by Problem 10-22 a), there exists a polynomial function $g$ of degree $n$, odd since $n-1$ is even, such that $g'=f$.

c) There exists a polynomial function $f$ of degree $n-1$, with $n-1$ being odd, and having only one root. Hence, $f(x)=0$ for one $x$. Moreover, by Problem 10-22 a), there exists a polynomial function $g$ of degree $n$, even since $n-1$ is odd, such that $g'=f$.

d) By Problem 7-4 a), there exists a polynomial function $f$ of degree $n-1$ with exactly $k$ roots such that $n-1-k$ is even. Hence, $f(x)=0$ for precisely $k$ numbers $x$. Moreover, by Problem 10-22 a), there exists a polynomial function $g$ of degree $n$ such that $g'=f$. In this case, it follows that $n-k$ is odd since $n-1-k$ is even.