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If -i is a zero of the polynomial k(x), then which of the following must be a factor of k(x) ?
C. X^2 -1
- Wayne DeguManLv 71 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)
- Jeff AaronLv 71 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.
- rotchmLv 71 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.
- llafferLv 71 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).