Factor completely x^3 + 6x^2 + 16x + 96
To factor the given expression, we can use synthetic division to find one of its roots.
Since the coefficient of the leading term is 1, the possible rational roots are factors of the constant term (96). The factors of 96 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±32, ±48, and ±96.
We can start testing the positive factors first, and if none work, we can test their negatives.
Testing x = 1:
1 | 1 6 16 96
| 1 7 23
1 7 23 119
Since the remainder is not zero, x - 1 is not a factor, and therefore 1 is not a root.
Testing x = 2:
2 | 1 6 16 96
| 2 16 64
1 8 32 160
Once again, x - 2 is not a factor, and therefore 2 is not a root.
Testing x = 3:
3 | 1 6 16 96
| 3 27 123
1 9 43 219
Similarly, x - 3 is not a factor, and therefore 3 is not a root.
Testing x = 4:
4 | 1 6 16 96
| 4 40 224
1 10 56 320
Once again, x - 4 is not a factor, and therefore 4 is not a root.
Testing x = 6:
6 | 1 6 16 96
| 6 72 456
1 12 88 552
Again, x - 6 is not a factor, and therefore 6 is not a root.
Testing x = 8:
8 | 1 6 16 96
| 8 112 864
1 14 128 960
Notably, x - 8 is a factor, and therefore 8 is a root.
Using synthetic division with x = 8:
8 | 1 6 16 96
| 8 112 864
1 14 128 960
The result of the division is 1x^2 + 14x + 128, which can be factored to be (x + 8)(x^2 + 6x + 16).
Therefore, the factored form of the expression x^3 + 6x^2 + 16x + 96 is (x + 8)(x^2 + 6x + 16).