Fill in the missing pieces in the factorization of x^3+2x^2−9x−18.
The factorization of x^3+2x^2−9x−18 is (x+3)(x+2)(x-3).
To fill in the missing pieces in the factorization of x^3+2x^2−9x−18, we can use a combination of synthetic division and factoring.
1. First, we need to find one factor of the polynomial. Since the polynomial has a constant term of -18, we can start by checking the factors of 18 (ignoring the negative sign for now). By trying different values, we find that x = 2 is a factor because when we substitute x = 2 into the polynomial, we get 0.
2. Once we have found one factor (x = 2), we can perform synthetic division to divide the polynomial by (x - 2):
```
2 │ 1 2 -9 -18
│ 2 8 -2
────────────────
1 4 -1 -20
```
The result of the synthetic division is 1x^2 + 4x - 1 as the quotient and -20 as the remainder.
3. Now, the polynomial can be factorized as follows:
x^3+2x^2−9x−18 = (x - 2)(x^2 + 4x - 1)
So, the missing pieces in the factorization are (x - 2)(x^2 + 4x - 1).