If `f(x) =x^4-8x^3+4x^2+4x+39 and f (3 + 2i) = a + ib` then `a : b` is equal to

Options

A. `1/8`
B. `-1/4`
C. `1/4`
D. `-1/8`

Answer

To solve the problem, we need to evaluate the function \( f(x) = x^4 - 8x^3 + 4x^2 + 4x + 39 \) at \( x = 3 + 2i \) and find the ratio \( a : b \) where \( f(3 + 2i) = a + ib \).

### Step-by-Step Solution:

1. **Substituting \( x = 3 + 2i \) into the function:**
\[ f(3 + 2i) = (3 + 2i)^4 - 8(3 + 2i)^3 + 4(3 + 2i)^2 + 4(3 + 2i) + 39 \]

2. **Calculating \( (3 + 2i)^2 \):**
\[ (3 + 2i)^2 = 3^2 + 2 \cdot 3 \cdot 2i + (2i)^2 = 9 + 12i - 4 = 5 + 12i \]

3. **Calculating \( (3 + 2i)^3 \):**
\[ (3 + 2i)^3 = (3 + 2i)(5 + 12i) = 3 \cdot 5 + 3 \cdot 12i + 2i \cdot 5 + 2i \cdot 12i \]
\[ = 15 + 36i + 10i - 24 = -9 + 46i \]

4. **Calculating \( (3 + 2i)^4 \):**
\[ (3 + 2i)^4 = (3 + 2i)(-9 + 46i) = 3 \cdot (-9) + 3 \cdot 46i + 2i \cdot (-9) + 2i \cdot 46i \]
\[ = -27 + 138i - 18i - 92 = -119 + 120i \]

5. **Substituting back into \( f(3 + 2i) \):**
\[ f(3 + 2i) = (-119 + 120i) - 8(-9 + 46i) + 4(5 + 12i) + 4(3 + 2i) + 39 \]

6. **Calculating each term:**
- \( -8(-9 + 46i) = 72 - 368i \)
- \( 4(5 + 12i) = 20 + 48i \)
- \( 4(3 + 2i) = 12 + 8i \)

7. **Combining all terms:**
\[ f(3 + 2i) = (-119 + 120i) + (72 - 368i) + (20 + 48i) + (12 + 8i) + 39 \]
\[ = (-119 + 72 + 20 + 12 + 39) + (120 - 368 + 48 + 8)i \]
\[ = (-119 + 72 + 20 + 12 + 39) + (-192)i \]
\[ = 24 - 192i \]

8. **Identifying \( a \) and \( b \):**
\[ a = 24, \quad b = -192 \]

9. **Finding the ratio \( a : b \):**
\[ a : b = 24 : -192 = \frac{24}{-192} = -\frac{1}{8} \]

### Final Answer:
The ratio \( a : b \) is \( -\frac{1}{8} \).