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

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

To find the coefficients of a cubic equation with given roots, we start by using the fact that if r is a root of a polynomial, then x-r is a factor of the polynomial. Therefore, the cubic equation with roots 3, 1, and -1 can be written as:

(x-3)(x-1)(x+1) = 0

Expanding the left-hand side, we get:

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

Simplifying, we get:

x^3 - 4x^2 + x - 3 = 0

Therefore, the coefficients of the cubic equation are 1, -4, 1, and -3. However, we can simplify this by dividing all the coefficients by the greatest common factor, which is 1 in this case. This gives us the final answer:

1x^3 - 3x^2 - 7x + 3 = 0

Therefore, the coefficients of the cubic equation with roots 3, 1, and -1 are 1, -3, -7, and 3.