Final answer:
To complete the synthetic division, use the factor theorem and the value of a. List the coefficients in descending order and perform the division. The result is the quotient.
Step-by-step explanation:
To complete the synthetic division, we will use the factor theorem which states that if (x - a) is a factor of a polynomial, then the remainder when the polynomial is divided by (x - a) is equal to zero. In this case, since (x - 1) is a factor, we will use 1 as the value of a in the synthetic division.
First, we list the coefficients of the polynomial in descending order: 1, 4, 1, -6. Then, we write down the value of a, which is 1, and perform the synthetic division:
(1) | 1 4 1 -6
| 1 5 6
+-----------
1 5 6 0
The result of the synthetic division is the quotient: 1x² + 5x + 6 = (x + 2)(x + 3). So, the polynomial factorization is y = (x - 1)(x + 2)(x + 3).