Question

Factor completely using synthetic division: Show all work, please!!!

f(x) = 2x^4 - 2x^3 - 3x^2 + 2x + 2

a) (2x^2 - 2x - 2)(x^2 + 1)
b) (2x - 1)(x^3 - 1)
c) (x^2 - 2)(2x^2 - 2)
d) (x - 1)(2x^3 + 2)

Answer 1

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).