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

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

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

Comparing this to the standard form of a cubic equation:

ax^3 + bx^2 + cx + d = 0

we can see that the coefficients are related to the roots by:

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

Substituting the given roots into these formulas, we get:

a = 1
b = -(2 + (-1) + 1) = -2
c = 2(-1)(1) + 2(2)(1) + (-1)(1) = -2
d = -(2)(-1)(1) = 2

Therefore, the coefficients of the cubic equation with roots 2, -1, and 1 are 1, -2, -3, and 2. The equation can be written as:

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