Asked by Anonymous
Given -4i is a root, determine all other roots of f(x) = x^3 - 3x^2 + 16x - 48.
Are they 4i and 3?
Are they 4i and 3?
Answers
Answered by
Reiny
complex roots come in conjugate pairs,
so we know two roots to be
-4i and 4i
two factors would be (x-4i) and (x+4i)
or (x^2 + 16) has to be a factor of f(x)
so it would need another linear factor.
since it ends in -48 and our x^2 factor end in +16, the remaining factor must end in -3
by this logic,
f(x) = (x-3)(x^2 + 16)
with roots of 3, 4i, and -4i
check: by Wolfram,
http://www.wolframalpha.com/input/?i=solve+x%5E3+-+3x%5E2+%2B+16x+-+48+%3D+0
so we know two roots to be
-4i and 4i
two factors would be (x-4i) and (x+4i)
or (x^2 + 16) has to be a factor of f(x)
so it would need another linear factor.
since it ends in -48 and our x^2 factor end in +16, the remaining factor must end in -3
by this logic,
f(x) = (x-3)(x^2 + 16)
with roots of 3, 4i, and -4i
check: by Wolfram,
http://www.wolframalpha.com/input/?i=solve+x%5E3+-+3x%5E2+%2B+16x+-+48+%3D+0
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