Grade 11Algebra
iz^3+z^2-z+i=0 ,then find out |z| value.plz ans. Of my question properly.
iz^3+z^2-z+i=0 ,then find out |z| value.plz ans. Of my question properly.
1 Answer
Arun
iz^3 + z^2 - z + i = 0
==> iz^2 (z - i) - 1(z - i) = 0, factoring by grouping
==> (iz^2 - 1) (z - i) = 0
==> i(z^2 + i) (z - i) = 0
==> z^2 + i = 0 or z - i = 0
==> z^2 = -i or z = i.
In either case, we have |z| = 1, because
(i) |z^2| = |-i| ==> |z|^2 = 1 ==> |z| = 1 (since |z| is non-negative real)
(ii) |z| = |i| ==> |z| = 1.
==> iz^2 (z - i) - 1(z - i) = 0, factoring by grouping
==> (iz^2 - 1) (z - i) = 0
==> i(z^2 + i) (z - i) = 0
==> z^2 + i = 0 or z - i = 0
==> z^2 = -i or z = i.
In either case, we have |z| = 1, because
(i) |z^2| = |-i| ==> |z|^2 = 1 ==> |z| = 1 (since |z| is non-negative real)
(ii) |z| = |i| ==> |z| = 1.
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