Final answer:
The expression x³-2x²-9x+18 is completely factored as (x-3)(x-2)(x+3).
Step-by-step explanation:
When factored completely, the expression x³-2x²-9x+18 is equivalent to (x-3)(x-2)(x+3). To factor the given cubic polynomial, we need to find the roots of the equation x³-2x²-9x+18 = 0. This can be done by trying out possible integer factors of the constant term, which is 18. Upon trial and error, we find that 3 is a root since substituting x=3 into the polynomial yields 0. Factoring by the root 3, we can perform polynomial long division or use synthetic division to get x²-5x+6. This resulting quadratic can be factored further to (x-2)(x-3). Combining this with our initial factor gives us the fully factored form of (x-3)(x-2)(x+3).