Question

write a polynomial of degree 6 in factored form with given zeros : sqrt(2), (1+i),3,-2. (rational values only

Answer 1

Great question! A polynomial with roots \(x=\left\{x_1,x_2,\ldots,x_n\right\}\) will have the factored form \((x-x_1)(x-x_2)\cdots(x-x_n)=0\) (think in reverse of how you find those roots in the first place). Keeping that in mind, a basic solution would be the following (yet not sufficient for this problem).\[(x-\sqrt 2)(x-(1+i))(x+2)(x-3)=0\]Since we are given the flexibility to express a polynomial of 6th degree, we will add additional factors as to collapse the current ones so that we get rational values only. Added factors have been underbraced.\[\underbrace{(x+\sqrt 2)}(x-\sqrt 2)\underbrace{(x-(1-i))}(x-(1+i))(x+2)(x-3)=0\]Distributing the extra terms through gives us the following as our final answer.\[\boxed{(x^2-2)(x^2-2x+2)(x+2)(x-3)=0}\]