Question

One root of a third degree polynomial function f(x) is –5 + 2i. Which statement describes the number and nature of all roots for this function? A. f(x) has two real roots and one complex root.
B. f(x) has two complex roots and one real root.
C. f(x) has three complex roots.
D. f(x) has three real roots.

Answer 1

Answer: Option 'B' is correct.

Step-by-step explanation:

Since we have given that

One root of a third degree polynomial function is -5+2i

As we know that every complex root comes with its conjugate root meaning they always come in pairs

So, another complex root will be -5-2i

So, remaining one root which must be real root because if it was complex it would come with its pair i.e. conjugate , then total it makes 4 roots which can't possible for three degree polynomial.

So, there must be 2 complex roots and one real root.

Hence, Option 'B' is correct.

Answer 2

Given the root of -5+2i, we are asked to describe the best statement that describes the polynomial. The root  -5+2i cannot exist without its conjugate root, -5-2i. Complex roots are always accompanied by at least 1 real root. Since there are only 3 roots, there should be one root only, hence the answer is B.