Question 51
Let $n$ be a positive integer, let $a_1, \dots, a_n$ be pairwise distinct real numbers, and let $b_1, \dots, b_n$ be arbitrary real numbers. Show that:
a) if the $b_i$ are all positive, then there exists a polynomial $f$ with real coefficients, no root of which is real, such that $f(a_i) = b_i$, $i = 1, \dots, n$;
b) there always exists a polynomial $f$ whose roots are all real, and $f(a_i) = b_i$, $i = 1, \dots, n$.
a) if the $b_i$ are all positive, then there exists a polynomial $f$ with real coefficients, no root of which is real, such that $f(a_i) = b_i$, $i = 1, \dots, n$;
b) there always exists a polynomial $f$ whose roots are all real, and $f(a_i) = b_i$, $i = 1, \dots, n$.