Question

The polynomial function f(x) is a fourth degree polynomial. Which of the following could be the complete list of the roots of f(x)?
3, 4, 5, 6
3, 4, 5, 6i
3, 4, 4 + i StartRoot 6 EndRoot, 5 + StartRoot 6 EndRoot
3, 4, 5 + i, –5 + i

Answer 1

If f has real coefficients (as is usual), nonreal roots must occur in conjugate pairs. Check each list:

- 3, 4, 5, 6 — all real, so this is possible (e.g. f(x) = (x−3)(x−4)(x−5)(x−6)).
- 3, 4, 5, 6i — 6i is nonreal but −6i is not listed, so impossible.
- 3, 4, 4 + i√6, 5 + √6 — 4 + i√6 is nonreal but its conjugate 4 − i√6 is missing, so impossible.
- 3, 4, 5 + i, −5 + i — neither nonreal root is the conjugate of the other, so impossible.

Therefore only the first list could be the complete set of roots.