Final Answer:
The polynomial f(x) = x⁴ + x^3 - 4x² - 2x + 4 factors completely into (x+1)(x-1)(x²+2).
Step-by-step explanation:
To factor the given polynomial f(x) = x⁴ + x³ - 4x² - 2x + 4 completely, we can use the rational root theorem to find the potential rational roots. The rational root theorem states that any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term (in this case, 4) and q is a factor of the leading coefficient (in this case, 1).
By testing these potential roots using synthetic division or polynomial long division, we find that x = -1 and x = 1 are roots of the polynomial. Therefore, we can factor out (x+1) and (x-1). After dividing f(x) by (x+1)(x-1), we get a quadratic expression x² + 2, which cannot be factored further over the real numbers. Thus, the complete factorization of f(x) is (x+1)(x-1)(x²+2).
Next, we can verify our factorization by multiplying the factors together to ensure they equal the original polynomial. Multiplying (x+1)(x-1)(x²+2) gives us x⁴ + x³ - 4x² - 2x + 4, which matches the original polynomial f(x). Therefore, our factorization is correct.
In conclusion, by using the rational root theorem and synthetic division, we found that the given polynomial f(x) = x⁴ + x³ - 4x² - 2x + 4 factors completely into (x+1)(x-1)(x²+2).