Three roots of a fifth degree polynomial function f(x) are –2, 2, and 4 + i. Which statement describes the number and nature of all roots for this function?

f(x) has two real roots and one imaginary root.
f(x) has three real roots.
f(x) has five real roots.
f(x) has three real roots and two imaginary roots.

Question

Three roots of a fifth degree polynomial function f(x) are –2, 2, and 4 + i. Which statement describes the number and nature of all roots for this function?

f(x) has two real roots and one imaginary root.
f(x) has three real roots.
f(x) has five real roots.
f(x) has three real roots and two imaginary roots.

campbell_st · Answer

well there are 2 imaginary roots, if you are given 4 + i the conjugate is also a root 4 - i

these roots come in pairs sometimes written

$$x = 4 \pm i$$

you are told 2 roots are real, -2, 2

a degree 5 polynomial has 5 roots.... 

so since imaginary roots come in pairs, the 5th root has to be real...

hope that helps

anonymous · Answer

So three real roots and two imaginary? Or would the all be considered "real"

campbell_st · Answer

your choice of 3 real 2 imaginary is the answer