Problem of the Week

Problem 36

Welcome back to Problem of the Week.

You may notice that there are two problems this week – Problem 36 is released a week early to provide a bit more time to solve. Going forward, the 1-week-1-problem structure will be mainly followed.

The solution to Problem 35 will be released on 7th October, and the solution to Problem 36 will be released on 14th October.

(a) Suppose $w = z - z^{-1}$ .
Express $z^2+z^{-2}$ in terms of $w$ .
Similarly express $z^3-z^{-3}$ in terms of $w$ .

(b) By considering division by $z^2$ , solve the quartic

(c*) Use a similar method in (b) to solve

$2z^8-3z^7-12z^6+12z^5+22z^4-12z^3-12z^2+3z+2=0$

(d*) Prove that the coefficients read the same forwards and backwards (the signs can be ignored for now) in a degree $8$ polynomial in $z$ if the polynomial has solutions in $w$ , $w=a,b,c,d$ .

(e*) Prove that if the coefficients read the same forwards and backwards and there are no solutions between $-1$ and $1$ inclusive, then there must not be any solutions in the degree $8$ polynomial in $z$ .