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

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 \( x^6 + x^4 + x^2 + 1 \) where \( x = 1 + i \).

### Step 1: Calculate \( x^2 \)
First, we 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 \( (a + b)^2 = a^2 + 2ab + b^2 \) and remember that \( i^2 = -1 \).

### Step 2: Calculate \( x^4 \)
Next, we calculate \( x^4 \) using \( x^2 \):
\[ x^4 = (x^2)^2 = (2i)^2 = 4i^2 = 4(-1) = -4 \]

**Hint:** Remember that squaring a complex number involves squaring the magnitude and doubling the angle, but here we just use \( i^2 = -1 \).

### Step 3: Calculate \( x^6 \)
Now, we calculate \( x^6 \) using \( x^2 \):
\[ x^6 = x^4 \cdot x^2 = (-4)(2i) = -8i \]

**Hint:** To find \( x^6 \), multiply \( x^4 \) and \( x^2 \) directly.

### Step 4: Combine all parts
Now we can substitute \( x^6 \), \( x^4 \), and \( x^2 \) into the expression:
\[ x^6 + x^4 + x^2 + 1 = (-8i) + (-4) + (2i) + 1 \]

### Step 5: Simplify the expression
Combine the real and imaginary parts:
\[ = (-4 + 1) + (-8i + 2i) = -3 - 6i \]

### Final Answer
Thus, the value of \( x^6 + x^4 + x^2 + 1 \) is:
\[ \boxed{-3 - 6i} \]