Answer:we cannot determine the complete factorization of the polynomial.
Explanation:
The polynomial x^3 + 7x^2 - 20x - 96 can be factored using synthetic division to determine if x - 4 is a factor.
To check if x - 4 is a factor, we divide the polynomial by x - 4 using synthetic division.
First, we set up the synthetic division like this:
4 | 1 7 -20 -96
|___________
Next, we bring down the first coefficient, which is 1:
4 | 1 7 -20 -96
|___________
1
Then, we multiply the divisor, 4, by the number at the bottom and write the result under the next coefficient:
4 | 1 7 -20 -96
|___________
1
4
Next, we add the numbers in the second column:
4 | 1 7 -20 -96
|___________
1
4
44
We repeat the process with the new number at the bottom, 44:
4 | 1 7 -20 -96
|___________
1
4
44
184
Lastly, we add the numbers in the third column:
4 | 1 7 -20 -96
|___________
1
4
44
184
736
The last number, 736, represents the remainder of the division.
If the remainder is 0, it means that x - 4 is a factor of the polynomial. However, since the remainder is not 0, x - 4 is not a factor of x^3 + 7x^2 - 20x - 96.
To completely factor the polynomial, we need to find its other factors. This can be done by factoring the polynomial further using methods like factoring by grouping or the rational root theorem. However, without any other information or constraints provided, we cannot determine the complete factorization of the polynomial.