Question 5
Let $a, b, c, d, e, f$ be real numbers such that the polynomial
$$
p(x) = x^{8} - 4x^{7} + 7x^{6} + a x^{5} + b x^{4} + c x^{3} + d x^{2} + e x + f
$$
factorises into eight linear factors $x - x_{i}$, with $x_{i} \gt 0$ for $i = 1, 2, \ldots, 8$. Determine all possible values of $f$.
$$
p(x) = x^{8} - 4x^{7} + 7x^{6} + a x^{5} + b x^{4} + c x^{3} + d x^{2} + e x + f
$$
factorises into eight linear factors $x - x_{i}$, with $x_{i} \gt 0$ for $i = 1, 2, \ldots, 8$. Determine all possible values of $f$.