To factor the polynomial [tex]\( x^2 - 8x + 16 \)[/tex] completely, we can follow these steps:
1. Identify the polynomial: We are given the polynomial [tex]\( x^2 - 8x + 16 \)[/tex].
2. Check if it fits the form of a perfect square trinomial: A perfect square trinomial takes the form [tex]\( a^2 - 2ab + b^2 \)[/tex] or [tex]\( a^2 + 2ab + b^2 \)[/tex]. We compare our polynomial to see if it matches this form.
In this case, we notice:
- The first term [tex]\( x^2 \)[/tex] is a perfect square ([tex]\( x^2 = (x)^2 \)[/tex]).
- The last term [tex]\( 16 \)[/tex] is also a perfect square ([tex]\( 16 = (4)^2 \)[/tex]).
- The middle term [tex]\(-8x\)[/tex] can be written as [tex]\(-2 \cdot x \cdot 4\)[/tex], which fits the form [tex]\(-2ab\)[/tex] with [tex]\( a = x \)[/tex] and [tex]\( b = 4 \)[/tex].
3. Rewrite the polynomial as a square of a binomial:
Given that it fits the form [tex]\( a^2 - 2ab + b^2 \)[/tex], we can write:
[tex]\[
x^2 - 8x + 16 = (x - 4)^2
\][/tex]
4. Verify the factorization:
To ensure correctness, we will expand [tex]\((x - 4)^2\)[/tex] and check if we get the original polynomial back:
[tex]\[
(x - 4)^2 = (x - 4)(x - 4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16
\][/tex]
This confirms that our factorization is correct.
Thus, the polynomial [tex]\( x^2 - 8x + 16 \)[/tex] factors completely as [tex]\( (x - 4)^2 \)[/tex], which can also be written as [tex]\( (x - 4)(x - 4) \)[/tex].
Hence, the correct answer is:
D. [tex]\( (x-4)(x-4) \)[/tex]
