Multiply the given binomials. Keep in mind that i^{2}=-1. Substitute when needed, write your answers in standard form.

1) 5i(-4+2i)
2) 3i(8-3i)
3) (2-i)(3+i)
4) (4+6i)(9-2i)



Let's go through each multiplication problem involving binomials, keeping in mind that \( i^2 = -1 \).

### 1) \( 5i(-4 + 2i) \)

Distribute \( 5i \) to both terms inside the parentheses:

\[
5i(-4 + 2i) = 5i(-4) + 5i(2i)
\]
\[
= -20i + 10i^2
\]

Since \( i^2 = -1 \), substitute \( -1 \) for \( i^2 \):

\[
= -20i + 10(-1)
\]
\[
= -20i - 10
\]

The standard form is:
\[
-10 - 20i
\]

### 2) \( 3i(8 - 3i) \)

Distribute \( 3i \) to both terms inside the parentheses:

\[
3i(8 - 3i) = 3i(8) + 3i(-3i)
\]
\[
= 24i - 9i^2
\]

Substitute \( i^2 = -1 \):

\[
= 24i - 9(-1)
\]
\[
= 24i + 9
\]

The standard form is:
\[
9 + 24i
\]

### 3) \( (2 - i)(3 + i) \)

Use the distributive property (FOIL: First, Outer, Inner, Last):

\[
(2 - i)(3 + i) = 2(3) + 2(i) - i(3) - i(i)
\]
\[
= 6 + 2i - 3i - i^2
\]

Simplify:

\[
= 6 - i - i^2
\]

Since \( i^2 = -1 \), substitute:

\[
= 6 - i - (-1)
\]
\[
= 6 - i + 1
\]
\[
= 7 - i
\]

The standard form is:
\[
7 - i
\]

### 4) \( (4 + 6i)(9 - 2i) \)

Use the distributive property (FOIL method):

\[
(4 + 6i)(9 - 2i) = 4(9) + 4(-2i) + 6i(9) + 6i(-2i)
\]
\[
= 36 - 8i + 54i - 12i^2
\]

Combine like terms:

\[
= 36 + 46i - 12i^2
\]

Substitute \( i^2 = -1 \):

\[
= 36 + 46i - 12(-1)
\]
\[
= 36 + 46i + 12
\]
\[
= 48 + 46i
\]

The standard form is:
\[
48 + 46i
\]

---

### Final answers in standard form:
1) \( -10 - 20i \)
2) \( 9 + 24i \)
3) \( 7 - i \)
4) \( 48 + 46i \)

---

Would you like further details on any of the steps?

Here are some additional related problems for practice:
1. Multiply \( (1 + i)(1 - i) \).
2. Multiply \( 7i(4 - 5i) \).
3. Simplify \( (3 + 2i)(2 - 4i) \).
4. Multiply \( (-6 + i)(5 - 3i) \).
5. Expand \( (1 + 3i)^2 \).

**Tip**: Always substitute \( i^2 = -1 \) early to simplify complex number expressions.

Multiply the given binomials. Keep in mind that i^{2}=-1. Substitute when needed, write your answers in standard form.
1) 5i(-4+2i)
2) 3i(8-3i)
3) (2-i)(3+i)
4) (4+6i)(9-2i)

Learn how to multiply complex binomials by applying the distributive property and the identity i^2 = -1. This guide offers step-by-step solutions to problems such as 5i(-4 + 2i), 3i(8 - 3i), (2 - i)(3 + i), and (4 + 6i)(9 - 2i). Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation