Question 23
Branko is writing a sequence of quadratic polynomials with real coefficients. In each step, he considers the polynomial he chose in the previous step, denoted by $ax^2+bx+c$, and proceeds to write down either $cx^2+bx+a$ or $a(x+d)^2+b(x+d)+c$ for some real number $d$.
Starting from $x^2-2x-1$ and performing a finite number of prescribed steps, can Branko obtain the following polynomial:
a) $2x^2 - 1$?
b) $2x^2 - x - 1$?
Starting from $x^2-2x-1$ and performing a finite number of prescribed steps, can Branko obtain the following polynomial:
a) $2x^2 - 1$?
b) $2x^2 - x - 1$?