If `z=3-4i` then `z^4-3z^3+3z^2+99z-95` is equal to

Options

A. 5
B. 6
C. `-5`
D. `-4`

Answer

To solve the expression \( z^4 - 3z^3 + 3z^2 + 99z - 95 \) for \( z = 3 - 4i \), we will follow these steps:

### Step 1: Substitute \( z \) into the expression
We start by substituting \( z = 3 - 4i \) into the polynomial:
\[ z^4 - 3z^3 + 3z^2 + 99z - 95 \]

### Step 2: Calculate \( z^2 \)
First, we calculate \( z^2 \):
\[ z^2 = (3 - 4i)^2 = 3^2 - 2 \cdot 3 \cdot 4i + (4i)^2 = 9 - 24i - 16 = -7 - 24i \]

### Step 3: Calculate \( z^3 \)
Next, we calculate \( z^3 \):
\[ z^3 = z \cdot z^2 = (3 - 4i)(-7 - 24i) = 3(-7) + 3(-24i) - 4i(-7) - 4i(-24i) \]
Calculating each term:
\[ = -21 - 72i + 28i + 96 = 75 - 44i \]

### Step 4: Calculate \( z^4 \)
Now, we calculate \( z^4 \):
\[ z^4 = z \cdot z^3 = (3 - 4i)(75 - 44i) = 3(75) + 3(-44i) - 4i(75) - 4i(-44i) \]
Calculating each term:
\[ = 225 - 132i - 300i - 176 = 49 - 432i \]

### Step 5: Substitute \( z^2, z^3, z^4 \) back into the polynomial
Now we substitute \( z^2, z^3, z^4 \) back into the expression:
\[ (49 - 432i) - 3(75 - 44i) + 3(-7 - 24i) + 99(3 - 4i) - 95 \]

### Step 6: Simplify the expression
Calculating each term:
\[ = (49 - 432i) - (225 - 132i) + (-21 - 72i) + (297 - 396i) - 95 \]

Combining the real parts:
\[ 49 - 225 - 21 + 297 - 95 = 5 \]

Combining the imaginary parts:
\[ -432i + 132i - 72i - 396i = -768i \]

### Final Result
Thus, the expression simplifies to:
\[ 5 - 768i \]

### Conclusion
Therefore, the value of \( z^4 - 3z^3 + 3z^2 + 99z - 95 \) is \( 5 - 768i \).

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