Solve the equation x4 − 4x2 + 8x + 35 = 0, if one of its roots is
2 + 3 i
Answers
Answered by
2
Since 2+i
3
is a root and the coefficients are real, the second root has to be 2−i
3
Thus, (x−2−i
3
)(x−2+i
3
)=x
2
−2x+ix
3
−2x+4−i2
3
−ix
3
+i2
3
+3
=x
2
−4x+7 is a root of the given equation.
Dividing the original equation by x
2
−4x+7, we have
(x
4
−4x
2
+8x+35)/(x
2
−4x+7)=x
2
+4x+5
Thus, we obtain the other root as x
2
+4x+5
Again factorizing this root, we get x=
2
−4±
16−20
=
2
−4±2i
=−2±i
The four roots are therefore 2+i
3
,2−i
3
,−2+i,−2−i
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