find the complex zeros of each polynomial function. Use your results to write the polynomial as a product of linear factors.

f(x) = x^4 + 6x^2 + 8

To find the complex zeros of the polynomial function \( f(x) = x^4 + 6x^2 + 8 \), we can start by making a substitution to simplify the equation.

### Step 1: Substitution

Let \( y = x^2 \). This turns the original equation into:

\[
f(y) = y^2 + 6y + 8
\]

This is now a quadratic equation in terms of \( y \).

### Step 2: Solve the quadratic equation

To solve the quadratic equation \( y^2 + 6y + 8 = 0 \), we can use the quadratic formula:

\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a = 1 \), \( b = 6 \), and \( c = 8 \).

\[
y = \frac{-6 \pm \sqrt{6^2 - 4(1)(8)}}{2(1)}
\]

\[
y = \frac{-6 \pm \sqrt{36 - 32}}{2}
\]

\[
y = \frac{-6 \pm \sqrt{4}}{2}
\]

\[
y = \frac{-6 \pm 2}{2}
\]

Thus, the two solutions for \( y \) are:

\[
y_1 = \frac{-6 + 2}{2} = -2, \quad y_2 = \frac{-6 - 2}{2} = -4
\]

### Step 3: Return to \( x^2 \)

Recall that \( y = x^2 \), so now we solve for \( x \) in each case:

1. \( x^2 = -2 \)  
   Taking the square root of both sides:
   \[
   x = \pm \sqrt{-2} = \pm i\sqrt{2}
   \]
   
2. \( x^2 = -4 \)  
   Taking the square root of both sides:
   \[
   x = \pm \sqrt{-4} = \pm 2i
   \]

### Step 4: Complex zeros

The complex zeros are:
\[
x = \pm i\sqrt{2}, \quad x = \pm 2i
\]

Thus, the complex zeros are:
\[
x = i\sqrt{2}, \quad x = -i\sqrt{2}, \quad x = 2i, \quad x = -2i
\]

### Step 5: Write the polynomial as a product of linear factors

The polynomial can be written as a product of linear factors based on its zeros:

\[
f(x) = (x - i\sqrt{2})(x + i\sqrt{2})(x - 2i)(x + 2i)
\]

This is the factorized form of the polynomial \( f(x) = x^4 + 6x^2 + 8 \).

---

Do you want more details or have any questions?

Here are some related questions for further practice:

1. Find the complex zeros of \( f(x) = x^4 + 5x^2 + 6 \).
2. Factor \( f(x) = x^4 + 2x^2 + 1 \) completely over the complex numbers.
3. Solve \( f(x) = x^4 + 4x^2 + 4 \) for its complex zeros.
4. Write \( f(x) = x^4 + 3x^2 + 2 \) as a product of linear factors.
5. Determine the complex roots of \( f(x) = x^4 + 7x^2 + 12 \).

**Tip:** When factoring higher-degree polynomials, look for ways to reduce them into simpler quadratic equations via substitution, especially when there is a clear pattern like \( x^4 \) and \( x^2 \).

Find the complex zeros of each polynomial function. Use your results to write the polynomial as a product of linear factors. f(x) = x^4 + 6x^2 + 8

Learn how to find the complex zeros of the polynomial f(x) = x^4 + 6x^2 + 8 using quadratic substitution and the quadratic formula. This step-by-step guide includes solving for complex numbers and writing the polynomial as a product of linear factors. Ideal for students in Grades 10-12.

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