Evaluating the Complex Number Expression \(x^6 + x^4 + x^2 + 1\) for \(x = 1 + i\)

We are given a complex number \(x = 1 + i\) and asked to find the value of the expression \(x^6 + x^4 + x^2 + 1\). To solve this, we need to calculate the powers of \(x\) individually and then substitute them into the expression.

Step-by-Step Calculation of Powers of \(x\)

First, let's find the value of \(x^2\):

\[x^2 = (1 + i)^2\]

Using the formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a=1\) and \(b=i\):

\[x^2 = 1^2 + 2(1)(i) + i^2\]

Recall that \(i^2 = -1\). Substituting this value:

\[x^2 = 1 + 2i - 1\]

\[x^2 = 2i\]

Next, let's find the value of \(x^4\). We can calculate this as \((x^2)^2\):

\[x^4 = (2i)^2\]

\[x^4 = 2^2 \cdot i^2\]

\[x^4 = 4 \cdot (-1)\]

\[x^4 = -4\]

Finally, let's find the value of \(x^6\). We can calculate this as \(x^2 \cdot x^4\):

\[x^6 = x^2 \cdot x^4\]

Substitute the values we found for \(x^2\) and \(x^4\):

\[x^6 = (2i) \cdot (-4)\]

\[x^6 = -8i\]

Evaluating the Expression \(x^6 + x^4 + x^2 + 1\)

Now we substitute the calculated values of \(x^6\), \(x^4\), and \(x^2\) into the given expression:

\[x^6 + x^4 + x^2 + 1 = (-8i) + (-4) + (2i) + 1\]

Rearrange the terms to group the real parts and the imaginary parts:

\[x^6 + x^4 + x^2 + 1 = (-4 + 1) + (-8i + 2i)\]

Combine the real parts:

\[-4 + 1 = -3\]

Combine the imaginary parts:

\[-8i + 2i = -6i\]

Putting the real and imaginary parts together, we get:

\[x^6 + x^4 + x^2 + 1 = -3 - 6i\]

Summary of Calculation

• Given \(x = 1 + i\)
• Calculated \(x^2 = 2i\)
• Calculated \(x^4 = -4\)
• Calculated \(x^6 = -8i\)
• Expression: \(x^6 + x^4 + x^2 + 1\)
• Substitute values: \((-8i) + (-4) + (2i) + 1\)
• Simplify: \((-4 + 1) + (-8i + 2i) = -3 - 6i\)

Thus, the value of the expression \(x^6 + x^4 + x^2 + 1\) when \(x = 1 + i\) is \(-3 - 6i\).