Forme algébrique (3) - Corrigé

Énoncé

Donner la forme algébrique de chacun des nombres complexes suivants.

\(z_1=2i-(8-i)\)

\(z_2=4i^2-2i+2(i-5)\)

\(z_3=4-i^3\)

\(z_4= (2+3i)(2-5i)\)

\(z_5=3i+\dfrac{5}{i}\)

\(z_6=(\dfrac{1+i}{3-i})^2\)

Solution

\(z_1=2i-(8-i)=-8+3i\)

\(z_2=4i^2-2i+2(i-5)=-4-2i+2i-10=-14\)

\(z_3=4-i^3=4-(-i)=4+i\)

\(z_4= (2+3i)(2-5i)=4-10i+6i+15=19-4i\)

\(z_5=3i+\dfrac{5}{i}=z_5=3i+\dfrac{5i}{i^2}=3i-5i=-2i\)

\(z_6=(\dfrac{1+i}{3-i})^2=\dfrac{1+2i-1}{9-6i-1}= \dfrac{2i}{8-6i} = \dfrac{2i(8+6i)}{100} = - \dfrac{3}{25}+i\dfrac{4}{25}\)