Question 734187
All third degree polynomials have three roots. If x=1 is a root, the polynomial can be factored so that (x-1) is a factor.<br>

If it is the only real root, and it's multiplicity one, that means that the other two roots are imaginary, and are also conjugates. The simplest pair of imaginary conjugate roots are i and -i. The quadratic equation with roots i and -i is (x+i)(x-i) = 0, or x^2+1 = 0.<br>

The third degree polynomial with 1, i and -i as roots is (x-1)(x^2+1) = 0, or x^3-x^2+x-1 = 0. There are other solutions, depending on what imaginary roots you use.<br>

If 1 has multiplicity three, that means that it is a triple root. The one and only one cubic equation with a root of 1 that has multiplicity three is (x-1)(x-1)(x-1) = 0, or x^3-3x^2+3x-1.