Let f(x)=xn +a1xn−1 +a2xn−2 +...+an,
where a1, a2, . . . , an are given numbers. It is given that f(x) can be written in the form
f (x) = (x + k1 )(x + k2 ) · · · (x + kn ). By considering f(0), or otherwise, show that k1k2 ...kn = an.
Show also that
(k1 +1)(k2 +1)···(kn +1)=1+a1 +a2 +...+an and give a corresponding result for (k1 − 1)(k2 − 1) · · · (kn − 1). (5)
Hint: For the last part, consider the cases n odd and n even separately.