Final answer:
To find the roots of the polynomial y = x^3 - 2x^2 - 3x + 6, we can first test possible rational roots and then use polynomial long division or synthetic division to find the remaining roots. This process reveals the roots as x = 2, x = -1, and x = -3, which corresponds to option (a).
Step-by-step explanation:
The roots or zeros of the polynomial y = x^3 - 2x^2 - 3x + 6 can be found by factoring the polynomial or using synthetic division. Since the options given indicate that the roots are likely integers, we can attempt to find them through factoring by inspection or testing possible rational roots. A common method to find roots is using the Rational Root Theorem, which suggests that possible roots are factors of the free term (6) divided by factors of the leading coefficient (which is 1 in this case).
By testing possible roots, we find that x = 2 is a root because substituting it into the polynomial gives:
y(2) = (2)^3 - 2(2)^2 - 3(2) + 6 = 8 - 8 - 6 + 6 = 0.
After finding one root, we can use synthetic division or long division to factor the cubic polynomial into a quadratic and a linear term. Factoring further or using the quadratic formula on the remaining quadratic expression will lead to the other roots.
Performing these steps, we arrive at the roots x = 2, x = -1, and x = -3, which matches option (a).