Prove that the polynomial x2x18n1 x is divisible by x21Solut

Prove that the polynomial (x^2+x+1)^(8n+1) -x is divisible by x^2+1.

Solution

Using the remainder theorem we can prove that (x^2+x+1)^(8n+1) -x is divisible by x^2+1.

x^2 + 1 = (x - i)(x + i)

f(x) = (x^2+x+1)^(8n+1) - x is divisible by x^2 + 1 if f(i) = 0 and f(-i) = 0

f(i) = (i^2 + i + 1)^(8n+1) - i

=> (-1 + i + 1)^(8n + 1) - i

=> i^8n*i - i

=> i*(i^8n - 1)

=> i*((-1)^4n - 1)

=> i*(1^2n - 1)

=> i*0

=> 0

f( -i)

=> ((-i)^2 - i + 1)^(8n+1) + i

=> (-1 - i + 1)^(8n + 1) + i

=> (-i)^(8n)*-i + i

=> -i*((-i)^8n - 1)

=> -i*((-1)^4n - 1)

=> -i*(1^2n - 1)

=> -i*0

=> 0

This has proved that (x^2+x+1)^(8n+1) - x is divisible by (x - i) and by (x + i)

Therefore (x^2+x+1)^(8n+1) - x is divisible by (x^2 + 1)