For the imaginary unit i, what is the simplified form of the expression (2 + 3i)(1 - 2i)? — ACT

For the imaginary unit $i$ , what is the simplified form of the expression $(2 + 3i)(1 - 2i)$ ?

8 - i

The correct answer is

8 - i

. To simplify the product of

(2 + 3i)(1 - 2i)

, we distribute the terms to get

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

. Since

i^2 = -1

, the last term

-6i^2

simplifies to

-6(-1) = 6

. Combining the real parts (

2 + 6 = 8

) and the imaginary parts (

-4i + 3i = -i

) yields

8 - i

Expand the product of the two binomials

(2 + 3i)(1 - 2i)

using the distributive property.

2 - 4i + 3i - 6i^2

Each term of the first binomial must be multiplied by each term of the second binomial.

Substitute

-1

for

i^2

in the expression.

2 - 4i + 3i - 6(-1) = 2 - 4i + 3i + 6

By definition, the imaginary unit

i

satisfies the equation

i^2 = -1

Combine the real terms and combine the imaginary terms to obtain the final simplified form.

8 - i

Adding the real parts (

2 + 6 = 8

) and the imaginary parts (

-4i + 3i = -i

) simplifies the expression to a single complex number.

Multiplying complex binomials and simplifying using the definition of the imaginary unit

i^2 = -1

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