For the complex number z = \frac{4 + 2i}{3 - i}, where i is the imaginary unit such that i… — ACT

For the complex number $z = \frac{4 + 2i}{3 - i}$ , where $i$ is the imaginary unit such that $i^2 = -1$ , what is the imaginary part of $z$ ?

The imaginary part of

z

1

To find the imaginary part of

z

, multiply the numerator and denominator of

\frac{4 + 2i}{3 - i}

by the complex conjugate of the denominator, which is

3 + i

. This simplifies the denominator to

9 - i^2 = 10

and the numerator to

12 + 10i + 2i^2 = 10 + 10i

. Dividing each term by

10

yields

1 + i

, which has an imaginary part of

1

Identify the complex conjugate of the denominator.

The complex conjugate of the denominator,

3 - i

, is

3 + i

To divide complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator to make the denominator a real number.

Multiply the numerator and denominator of the fraction by the complex conjugate.

z = \frac{(4 + 2i)(3 + i)}{(3 - i)(3 + i)}

Multiplying by

\frac{3 + i}{3 + i}

is equivalent to multiplying by

1

, which preserves the value of the complex number.

Expand and simplify the numerator and denominator.

Numerator:

(4 + 2i)(3 + i) = 12 + 4i + 6i + 2i^2 = 12 + 10i + 2(-1) = 10 + 10i

. Denominator:

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

Apply the distributive property (FOIL method) and substitute

i^2 = -1

to simplify the expression.

Divide each term in the numerator by the denominator to find the imaginary part.

z = \frac{10 + 10i}{10} = 1 + i

, so the imaginary part is

1

Rewrite the fraction in standard form

a + bi

, where the imaginary part is

b

(the coefficient of

i

Simplifying a quotient of complex numbers by multiplying both the numerator and the denominator by the complex conjugate of the denominator.

Soru