The polynomial p(x) = x5 + x + 2 has A. all real roots B. 3 real and 2 complex roots C. 1 real and 4 complex roots D. all complex roots

The correct answer is: A. all real roots.

A polynomial with all real roots must have an odd degree. This is because the sum of the roots of a polynomial is always equal to its constant term, and the product of the roots is always equal to its leading coefficient. In this case, the constant term is 2 and the leading coefficient is 1, so the sum of the roots must be 2 and the product of the roots must be 1.

Since the sum of the roots is real, at least one of the roots must be real. If one of the roots is real, then the product of the other roots must also be real, since the product of a real number and an imaginary number is always imaginary. Therefore, all of the roots must be real.

Here is a brief explanation of each option:

• Option A: all real roots. This is the correct answer, as explained above.
• Option B: 3 real and 2 complex roots. This is not possible, as the polynomial has an odd degree.
• Option C: 1 real and 4 complex roots. This is not possible, as the polynomial has an odd degree.
• Option D: all complex roots. This is not possible, as the polynomial has an odd degree.