For the imaginary unit i, where i^2 = -1, the complex number z is defined as z = (3 - 2i)^… — ACT

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = (3 - 2i)^2 - 4i^{103}$ . What is the imaginary part of $z$ ?

Answer: -8

Answer

The imaginary part of

z

-8

The expression

(3-2i)^2

expands to

9 - 12i + 4i^2 = 5 - 12i

. The term

i^{103}

simplifies to

-i

since

103

leaves a remainder of

3

when divided by

4

. Subtracting

4i^{103}

corresponds to adding

4i

, giving

z = (5-12i) + 4i = 5-8i

. The coefficient of the imaginary part is

-8

Step-by-Step Solution

Expand

(3 - 2i)^2

5 - 12i

Use the binomial expansion formula and substitute

i^2 = -1

Simplify the term

-4i^{103}

4i

Since

103

divided by

4

leaves a remainder of

3

i^{103} = i^3 = -i

. Therefore,

-4i^{103} = -4(-i) = 4i

Combine terms to find

z

and identify its imaginary part

-8

Add the components:

z = (5 - 12i) + 4i = 5 - 8i

. The imaginary part is the coefficient of

i

, which is

-8

Key Concept

Operations on complex numbers including binomial expansion, powers of the imaginary unit

i

, and identification of the imaginary part.

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