**What is the least possible degree of a polynomial that has roots -5,1 + 4i, and -4i?**

a. 3

b. 2

c. 5

d. 4

**Answer:**

c. 5

Without any extra conditions, the answer could be 3, and the simplest polynomial with the given roots would be

(x + 5) (x – (1 + 4i )) (x + 4i )

= x ³ + 4x ² + (11 – 4i ) x + 80 – 2i

If the polynomial is supposed to have only real coefficients, then any complex roots must occur along with their complex conjugates:

(x + 5) (x – (1 + 4i )) (x – (1 – 4i )) (x + 4i ) (x – 4i )

= x ⁵ + 3x ⁴ + 23x ³ + 133x ² + 112x + 1360

and then the degree would be 5.