The expression given is \( x^4 - 8x^2 + 16 \). This is a polynomial expression, and it can be factored to simplify it further.

### Factoring the Polynomial

Notice that the expression can be seen as a quadratic in terms of \( x^2 \):

\[
x^4 - 8x^2 + 16 = (x^2)^2 - 8(x^2) + 16
\]

We can treat \( x^2 \) as a single variable, say \( y \), so the expression becomes:

\[
y^2 - 8y + 16
\]

This is a simple quadratic equation, and we can factor it as:

\[
y^2 - 8y + 16 = (y - 4)(y - 4) = (y - 4)^2
\]

Now, substitute back \( y = x^2 \):

\[
(x^2 - 4)^2
\]

Finally, we can factor \( x^2 - 4 \) further as a difference of squares:

\[
x^2 - 4 = (x - 2)(x + 2)
\]

So, the fully factored form of the expression is:

\[
(x - 2)^2(x + 2)^2
\]

### Summary

The factored form of \( x^4 - 8x^2 + 16 \) is:

\[
(x - 2)^2(x + 2)^2
\]

Would you like any further details or explanations?

Here are five related questions to explore this topic further:

1. How can you factor other quadratic expressions?
2. What is the difference of squares, and how can it be used in factoring?
3. How do you solve polynomial equations using the factored form?
4. What are the roots of the equation \( x^4 - 8x^2 + 16 = 0 \)?
5. How do the graphs of quadratic and quartic functions differ?

**Tip:** When factoring, always look for common patterns like the difference of squares, perfect square trinomials, or substitution to simplify the process.

Discover how to factor the polynomial expression x^4 - 8x^2 + 16 with a clear, step-by-step explanation. This problem highlights essential algebraic techniques such as substitution and the difference of squares, making it ideal for students in Grades 10-12.

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