Final answer:
The polynomial equation has complex roots ±3 i, and by using the conjugate root theorem along with the relationship between the product of the roots and the constant term of the polynomial, the other root of the equation is found to be -1.
Step-by-step explanation:
The polynomial equation given is x^3 + x^2 + 9x + 9 = 0. Since it's given that the equation has complex roots ±3 i, by the conjugate root theorem, we know that complex roots come in conjugate pairs, so both 3i and -3i are roots of the equation. Therefore, the equation can be written as (x - 3i)(x + 3i)(x - a) = 0, where a is the unknown real root that we are looking to find.
We can use the fact that the product of the roots of a polynomial equals the constant term (in this case 9) divided by the leading coefficient (which is 1). Therefore a * (-3i) * (3i) = 9, which simplifies to a = -1.
We can also verify this by expanding (x - 3i)(x + 3i) to get x^2 + 9, and then our equation becomes (x^2 + 9)(x - a) = 0. Setting x^2+9 to zero gives us the complex roots we started with, so setting x - a to zero gives us the real root, a.