Final answer:
To factor the polynomial completely using synthetic division, we divide it by the possible factors and check for a remainder of zero. In this case, (x - 1) and (x + 1) are factors of the polynomial. By performing the division, we find the final factored form: (x - 1)(x + 1)(2x^2 - 2) = (2x^2 - 2)(x^2 - 1).
Step-by-step explanation:
To factor the polynomial completely using synthetic division, we need to find its roots. We start by trying possible factors using synthetic division and check if any of them result in a remainder of zero. Considering the polynomial f(x) = 2x^4 - 2x^3 - 3x^2 + 2x + 2, we can divide it by x - 1 and x + 1 to get a remainder of zero in both cases. This means that (x - 1) and (x + 1) are factors of f(x).
By dividing f(x) by (x - 1) and (x + 1) using synthetic division, we get the quotient: 2x^2 - 2 and the remainder: 2x + 4. Therefore, f(x) = (x - 1)(x + 1)(2x^2 - 2) = (x^2 - 1)(2x^2 - 2) = (2x^2 - 2)(x^2 - 1).