If x =1 +i, then the value of `x^(6)+x^(4)+x^(2)+1` is

Options

A. 6i-3
B. `-6i+3`
C. `-6i-3`
D. `6i+3`

Answer

To solve the problem, we need to find the value of \( x^6 + x^4 + x^2 + 1 \) given that \( x = 1 + i \).

### Step-by-Step Solution:

1. **Calculate \( x^2 \)**:
\[ x^2 = (1 + i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i + (-1) = 2i \]
**Hint**: Use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) to expand \( (1 + i)^2 \).

2. **Calculate \( x^4 \)**:
\[ x^4 = (x^2)^2 = (2i)^2 = 4i^2 = 4(-1) = -4 \]
**Hint**: Remember that \( i^2 = -1 \).

3. **Calculate \( x^6 \)**:
\[ x^6 = x^4 \cdot x^2 = (-4) \cdot (2i) = -8i \]
**Hint**: Multiply the results of \( x^4 \) and \( x^2 \) directly.

4. **Substitute into the expression \( x^6 + x^4 + x^2 + 1 \)**:
\[ x^6 + x^4 + x^2 + 1 = (-8i) + (-4) + (2i) + 1 \]
**Hint**: Combine like terms carefully, separating real and imaginary parts.

5. **Combine the terms**:
\[ = (-8i + 2i) + (-4 + 1) = -6i - 3 \]
**Hint**: Group the imaginary parts together and the real parts together.

### Final Answer:
The value of \( x^6 + x^4 + x^2 + 1 \) is \( -6i - 3 \).