If `z = -5 + 3i`, find the value of `(z^(4)+9z^(3)+26z^(2)-14z + 8)`.

Options

A. `60`
B. `-60`
C. `-48`
D. `48`

Explanation

To solve the expression \( z^4 + 9z^3 + 26z^2 - 14z + 8 \) where \( z = -5 + 3i \), we will calculate each power of \( z \) step by step.

### Step 1: Calculate \( z^2 \)

\[ z^2 = (-5 + 3i)^2 = (-5)^2 + 2(-5)(3i) + (3i)^2 = 25 - 30i - 9 = 16 - 30i \]

**Hint:** Use the formula \( (a + bi)^2 = a^2 + 2abi + (bi)^2 \) to expand the square.

### Step 2: Calculate \( z^3 \)

\[ z^3 = z \cdot z^2 = (-5 + 3i)(16 - 30i) \]
\[ = -5 \cdot 16 + (-5)(-30i) + 3i \cdot 16 + 3i \cdot (-30i) \]
\[ = -80 + 150i + 48i + 90 = 10 + 198i \]

**Hint:** Multiply the complex numbers using the distributive property and remember that \( i^2 = -1 \).

### Step 3: Calculate \( z^4 \)

\[ z^4 = z \cdot z^3 = (-5 + 3i)(10 + 198i) \]
\[ = -5 \cdot 10 + (-5)(198i) + 3i \cdot 10 + 3i \cdot 198i \]
\[ = -50 - 990i + 30i + 594 = 544 - 960i \]

**Hint:** Again, use the distributive property to multiply the complex numbers.

### Step 4: Substitute into the expression

Now we substitute \( z^4 \), \( z^3 \), \( z^2 \), and \( z \) into the expression \( z^4 + 9z^3 + 26z^2 - 14z + 8 \):

\[ = (544 - 960i) + 9(10 + 198i) + 26(16 - 30i) - 14(-5 + 3i) + 8 \]

Calculating each term:

1. \( 9z^3 = 9(10 + 198i) = 90 + 1782i \)
2. \( 26z^2 = 26(16 - 30i) = 416 - 780i \)
3. \( -14z = -14(-5 + 3i) = 70 - 42i \)

Now combine all the terms:

\[ = (544 + 90 + 416 + 70 + 8) + (-960i + 1782i - 780i - 42i) \]

Calculating the real part:

\[ 544 + 90 + 416 + 70 + 8 = 1128 \]

Calculating the imaginary part:

\[ -960 + 1782 - 780 - 42 = 0 \]

### Final Result

Thus, the value of \( z^4 + 9z^3 + 26z^2 - 14z + 8 \) is:

\[ \boxed{1128} \]