Question

Suppose that R(x) is a polynomial of degree 7 whose coefficients are real numbers.

Answer 1

R(x) is a polynomial of degree 7, so it has up to 7 distinct complex roots r₁, ..., r₇, and we can write it in terms of these roots as

R(x) = (x - r₁) (x - r₂) ... (x - r₇)

The coefficients of R(x) are all real, so the roots must all be complex numbers, and any of these roots with non-zero imaginary parts must occur along with their complex conjugates. This means if a + b i is a root, then is a - b i is also a root.

(a) We're told that -5 - 3i and 2i are roots to R(x), so we also know that -5 + 3i and -2i are roots.

There are 4 roots accounted for, leaving us with 3 unknown roots. These roots cannot all be non-real, because we can only count 2 of them as a conjugate pair. So we can have either

(b) at most 3 real roots, or

(c) at most 2 non-real roots and 1 real root.