Find all 4 roots of the polynomial: f ( x ) = x 4 </msup> + 2 x

zintekensci 2022-05-23 Answered
Find all 4 roots of the polynomial:
f ( x ) = x 4 + 2 x 3 x 1.
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Answers (1)

zepplinkid7yk
Answered 2022-05-24 Author has 11 answers
With the substitution x = t b / n a, you eliminate the second highest order term of a polynomial a x n + b x n 1 + .
If you substitute x = t 2 / 4 = t 1 / 2 your polynomial becomes
t 4 3 2 t 2 11 16 .
This is quadratic in t 2 so you can solve to get
t 2 = 3 ± 2 5 4 .
So the four roots of your polynomial are plus and minus the square roots of those two solutions (plus 1/2 because of the original substitution.) We were lucky that the first order term also disappeared.
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