Questions: If A=-3+5i, B=4-2i, and C=1+6i, where i is the imaginary unit, then A-BC equals: 1) 5-17i (-3+5i)-(4-2i)(1+6i) 2) 5+27i 3) -19-17i 4) -19+27i

Transcript text: 2) If $A=-3+5 i, B=4-2 i$, and $C=1+6 i$, where $i$ is the imaginary unit, then $A-B C$ equals: 1) $5-17 i$ \[ (-3+5 i)-(4-2 i)(1+6 i) \] 2) $5+27 i$ 3) $-19-17 i$ 4) $-19+27 i$

Solution

Solution Steps

To solve the problem, we need to compute the expression $ A - BC $ where $ A = -3 + 5i $, $ B = 4 - 2i $, and $ C = 1 + 6i $. First, calculate the product $ BC $ by multiplying the complex numbers $ B $ and $ C $. Then, subtract the result from $ A $.

Step 1: Calculate $ BC $

To find $ BC $, we multiply the complex numbers $ B $ and $ C $: \[ B = 4 - 2i, \quad C = 1 + 6i \] Using the formula for multiplying complex numbers: \[ BC = (4)(1) + (4)(6i) + (-2i)(1) + (-2i)(6i) = 4 + 24i - 2i + 12 = 16 + 22i \]

Step 2: Calculate $ A - BC $

Now, we subtract $ BC $ from $ A $: \[ A = -3 + 5i \] Thus, \[ A - BC = (-3 + 5i) - (16 + 22i) = -3 - 16 + (5i - 22i) = -19 - 17i \]