Final answer:
A fourth degree polynomial with real coefficients and roots at −2, could also have complex conjugate roots at 1 + √2 ± i.
Step-by-step explanation:
The question is asking about the possible roots of a fourth degree polynomial with real coefficients. Given that one of the roots is −2 and there are complex roots 1 + √2 ± i, we must remember that non-real roots of polynomials with real coefficients come in conjugate pairs.
This means that if 1 + √2 + i is a root, its conjugate 1 + √2 - i must also be a root.
Thus, the complete set of roots for this fourth degree polynomial can be −2, 1 + √2 + i, and 1 + √2 - i. Since we have identified all four roots (including the complex conjugate pair), these must be all the roots of the polynomial. It is important to note that while solving quadratic equations, which are second-order polynomials, the concept of complex conjugate roots also applies.
However, in physical problems, quadratic equations based on physical data will always result in real roots, often with only the positive roots being significant.