Answer:
The polynomial f(x) = x^4 + 2x^3 - 6x^2 + 4x - 16 is irreducible over the rational numbers.
None of the option is true.
Explanation:
To factor the polynomial function f(x) = x^4 + 2x^3 - 6x^2 + 4x - 16, we can attempt to find its roots and then factorize accordingly using synthetic division or other methods.
Upon inspection, we can see that there are no linear factors (roots) for this polynomial by testing values such as 1, -1, 2, -2, etc.
Next, we can check for possible quadratic factors. We can try to factorize by using the rational root theorem.
According to the theorem, any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term (-16) and q is a factor of the leading coefficient (1).
The factors of 16 are ±1, ±2, ±4, ±8, ±16, and the factors of 1 are ±1.
However, after testing these possible rational roots, we find that none of them are roots of the polynomial f(x).
Hence, we conclude that the polynomial does not factorize into linear or quadratic factors with rational roots.
Therefore,
The polynomial is irreducible over the rational numbers.
None of the option is true.