Factor the Quadratic Polynomial: Simplifying x² - 16x + 64

To find the product representation of the function $f(x) = x^2 - 16x + 64$, we expect it to be a perfect square trinomial.

First, recognize that the given quadratic form is $a^2 - 2ab + b^2$. Comparing it with $x^2 - 16x + 64$:

• The first term $x^2$ suggests $a = x$.
• The middle term $-16x$ can be interpreted as $-2ab$. So, $-2a = -16$, solving gives $2b = 16$, hence $b = 8$.
• The last term $(64)$ is $b^2$, confirms that $8^2 = 64$.

This means our expression is:

• $f(x) = (x - 8)^2$

Thus, the product form or factored representation of the function is $(x-8)^2$.

The final answer is: $(x-8)^2$.