If f element integers2 x prove that fx2 fx2 You may use 1 t

If f element integers_2 [x], prove that f(x^2) = (f(x))^2. You may use (1) to prove that if alpha is a root of f(x), then alpha^2 will also be a root of f(x).

Solution

Let f(x) =a[0]+a[1]x+..........a[n]xⁿ (a[i] in Z₂)

f(x)² (by Multinomial theorem)

= a[0]² + a[1]² x² +..........+a[n]² x²ⁿ + 2(cross terms)

=a[0] + a[1]x² +..........+a[n]x²ⁿ (as 0² =0 and 1² =1)

=f(x²)

(2) If a is a root of f(x) , then f(a²) = f(a)² (from above)=0. So a² is a root of f(x)

as required.