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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 

4 Answers

  • 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).

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