Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

If -i is a zero of the polynomial k(x), then which of the following must be a factor of k(x) ?

A. i

B. X^2ix+1

C. X^2 -1

D. X^2+1

E. X^2

Relevance
• 1 month ago

If x = -i is a root so is x = i

so, x = ±i

Hence, x² = -1

i.e. x² + 1 = 0

so, x² + 1 is a quadratic factor of k(x)

:)>

• 1 month ago

(x + i)

If k(x) has all real coefficients, then (x - i) must also be a root, so therefore x^2 - i^2, which is x^2 + 1, is also a root. That's similar to answer D, except for some reason answer D uses a capital X, whereas the question uses a small x.

But if k(x) can have non-real coefficients, then (x + i) is the only factor we know for sure.

• rotchm
Lv 7
1 month ago

Hint: then its conjugate is a root.

Don't forget to vote me best answer for being the first to hint you through without spoiling the answer. That way it gives you a chance to work at it and to get good at it.

If you were not Anonymous, I would hint further.

• 1 month ago

If -i is a root then "i" is also a root (its conjugate).

With two roots known, we can turn them into factors by subtracting them from "x":

(x - i) and (x + i)

The product of two factors will also be a factor, so:

(x - i)(x + i)

x² + xi - xi - i²

x² - i²

i² = -1, so:

x² + 1

This (answer D) will be a factor of k(x).