I must find 3rd degree Polynomial functions in R[x] with:
1) no roots
2) only one root
3) only two roots
4) only 3 roots
If the function has a root, then prove it. If not, then explain why.
My attempt:
We know, that the cubic function can have one, two or three roots.
But I really don't know, how I can find the polynomial functions.
1) Explanation
A 3rd polynomial function can not have no root because a polynomial function have at least one root. (Continous function)
Best Answer
Hint: make them in a standard form as $(x-x_1)(x-x_2)(x-x_3)$ where $x_1,x_2,x_3$ are the roots. For example for the case where we have $2$ zero roots we have $$x_1=x_2=0$$and choosing $x_3=1$ arbitrarily we obtain$$f(x)=x^2(x-1)$$