3. Given that $\displaystyle f(x)=x^n-nx+n-1$ for integer n>1. By considering f(x) and f '(x), show that $\displaystyle f(x)=(x-1)^2 g(x)$ is true for all polynomials g(x) with integer coefficients. Hence or otherwise

a) Show that $\displaystyle 3^{2n}-8n-1$ is divisible by 64 for all integers n>1

b) Show that the equation $\displaystyle x^4-4x+3=0$ does not have any real root other than 1

4. If $\displaystyle 3+\sqrt{3}$ is a root of the equation $\displaystyle 2x^2+3bx+3c=0$ where b and c are constants. Find the value of bc.

Answer:

4. bc=-16