As a complex number, what is the product of (i^3 - i^2)(3i^4 + i)?

As a complex number, what is the product of (i^3 - i^2)(3i^4 + i)?

• A. 2 + 4i
• B. 4 - 2i
• C. -4 + 2i
• D. -2 - 4i

Answer: (B)

To find the product of the expression (i^3 - i^2)(3i^4 + i), we first need to simplify each part. Starting with i^3 and i^2, we know the powers of the imaginary unit i follow a pattern: - i^1 = i - i^2 = -1 - i^3 = -i - i^4 = 1 Using this, we can simplify i^3 - i^2: - i^3 = -i - i^2 = -1 Thus: i^3 - i^2 = -i - (-1) = -i + 1 = 1 - i. Next, we simplify 3i^4 + i: - i^4 = 1 (as established), so: 3i^4 = 3 * 1 = 3. Therefore: 3i^4 + i = 3 + i. Now we can multiply the two expressions we simplified: (1 - i)(3 + i). Using the distributive property (also known as the FOIL method for binomials), we calculate: 1 * 3 = 3, 1 * i = i,

To find the product of the expression (i^3 - i^2)(3i^4 + i), we first need to simplify each part.

Starting with i^3 and i^2, we know the powers of the imaginary unit i follow a pattern:

• i^1 = i

• i^2 = -1

• i^3 = -i

• i^4 = 1

Using this, we can simplify i^3 - i^2:

• i^3 = -i

• i^2 = -1

Thus:

i^3 - i^2 = -i - (-1) = -i + 1 = 1 - i.

Next, we simplify 3i^4 + i:

• i^4 = 1 (as established), so:

3i^4 = 3 * 1 = 3.

Therefore:

3i^4 + i = 3 + i.

Now we can multiply the two expressions we simplified:

(1 - i)(3 + i).

Using the distributive property (also known as the FOIL method for binomials), we calculate:

1 * 3 = 3,

1 * i = i,