3x^(4)-11x^(3)+15x^(2)-9x+2

Question

Find all of the roots

Answer 1

By inspection x=1 is a root, so divide by this root:

1 | 3 -11 15 -9  2

     3   3  -8   7 -2

     3  -8   7 -2 | 0 = 3x^3-8x^2+7x-2, which, by inspection, also has x=1 as a root, so divide again:

1 | 3 -8  7 -2

     3  3 -5  2

     3 -5  2 | 0 = 3x^2-5x+2=(x-1)(3x-2).

The roots are 1 (three times) and 2/3: factorisation is (x-1)^3(3x-2).