Determine the coefficients of the cubic equation with roots 1, -2, and 3.

The cubic equation with roots 1, -2, and 3 has coefficients -6, 2, -7, and 1.

To find the coefficients of a cubic equation with given roots, we start with the factored form of the equation:

(x - r1)(x - r2)(x - r3) = 0

where r1, r2, and r3 are the roots. Expanding this expression gives:

x^3 - (r1 + r2 + r3)x^2 + (r1r2 + r1r3 + r2r3)x - r1r2r3 = 0

Therefore, the coefficients of the cubic equation are:

a = -1
b = r1 + r2 + r3
c = -(r1r2 + r1r3 + r2r3)
d = r1r2r3

Substituting the given roots, we get:

a = -1
b = 1 + (-2) + 3 = 2
c = -[(1)(-2) + (1)(3) + (-2)(3)] = -(-2 + 3 - 6) = 5
d = (1)(-2)(3) = -6

Therefore, the cubic equation with roots 1, -2, and 3 is:

x^3 + 2x^2 + 5x - 6 = 0