Complex Arithmetic/Examples/z 1^3 - 3 z 1^2 + 4 z 1 - 8

Example of Complex Arithmetic

Let $z_1 = 2 + i$.

Then:

$z_1^3 - 3 z_1^2 + 4 z_1 - 8 = -7 + 3 i$

Proof

\(\ds z_1^2\)

\(=\)

\(\ds \paren {2 + i} \paren {2 + i}\)

\(\ds \)

\(=\)

\(\ds 4 + 4 i + i^2\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds 3 + 4 i\)

\(\ds z_1^3\)

\(=\)

\(\ds z_1^2 \times z_1\)

\(\ds \)

\(=\)

\(\ds \paren {3 + 4 i} \paren {2 + i}\)

\(\ds \)

\(=\)

\(\ds 6 + 8 i + 3 i + 4 i^2\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds 2 + 11 i\)

So:

\(\ds z_1^3 - 3 z_1^2 + 4 z_1 - 8\)

\(=\)

\(\ds \paren {2 + 11 i} - 3 \paren {3 + 4 i} + 4 \paren {2 + i} - 8\)

\(\ds \)

\(=\)

\(\ds \paren {2 + 11 i} - \paren {9 + 12 i} + \paren {8 + 4 4} - 8\)

\(\ds \)

\(=\)

\(\ds \paren {2 - 9 + 8 - 8} + \paren {11 - 12 + 4} i\)

\(\ds \)

\(=\)

\(\ds -7 + 3 i\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $2 \ \text{(b)}$