Let i be the imaginary unit such that i^2 = -1. What is the simplified form of the express… — ACT

Let $i$ be the imaginary unit such that $i^2 = -1$ . What is the simplified form of the expression $(1 + 2i)^2(3 - i)$ ?

-5 + 15i

The correct answer is

-5 + 15i

. We first expand the squared binomial

(1 + 2i)^2

, which results in

1 + 4i + 4i^2

. Substituting

i^2 = -1

yields

-3 + 4i

. We then multiply this by the second binomial

(3 - i)

to get

-9 + 3i + 12i - 4i^2

. Substituting

i^2 = -1

one more time and combining like terms leads to the final simplified result of

-5 + 15i

Expand the squared binomial

(1 + 2i)^2

1 + 4i + 4i^2

Before multiplying by the second binomial, we must apply the exponent to the first binomial according to the order of operations.

Substitute

i^2 = -1

to simplify the expression from Step 1

-3 + 4i

Since

i^2 = -1

, the term

4i^2

becomes

4(-1) = -4

, and combining the real parts gives

1 - 4 = -3

Multiply the simplified term by

(3 - i)

using the FOIL method

-9 + 3i + 12i - 4i^2

We distribute each term of the first binomial into the second binomial:

(-3)(3) = -9

(-3)(-i) = 3i

(4i)(3) = 12i

, and

(4i)(-i) = -4i^2

Simplify the resulting expression by combining like terms and substituting

i^2 = -1

-5 + 15i

Combining the imaginary parts gives

3i + 12i = 15i

. Substituting

i^2 = -1

into

-4i^2

gives

-4(-1) = +4

. Finally, combining the real parts yields

-9 + 4 = -5

Simplification of complex expressions involving binomial squaring and multiplication under the definition

i^2 = -1

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