To factor the polynomial [tex]\( z^2 - 10z + 25 \)[/tex]:
1. Identify the polynomial structure: It is a quadratic polynomial of the form [tex]\( az^2 + bz + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 25 \)[/tex].
2. Check if it can be factored by recognizing perfect squares:
- A perfect square trinomial takes the form [tex]\( (z - p)^2 \)[/tex], which expands to [tex]\( z^2 - 2pz + p^2 \)[/tex].
- Here, compare the given polynomial [tex]\( z^2 - 10z + 25 \)[/tex] to [tex]\( z^2 - 2pz + p^2 \)[/tex]:
[tex]\[
z^2 - 10z + 25 = z^2 - 2 \cdot 5 \cdot z + 5^2
\][/tex]
So, you can see that [tex]\( p = 5 \)[/tex].
3. Write the factors: Since [tex]\( p = 5 \)[/tex], the polynomial can be factored as:
[tex]\[
(z - 5)^2
\][/tex]
So, the factored form of the polynomial [tex]\( z^2 - 10z + 25 \)[/tex] is:
[tex]\[
(z - 5)^2
\][/tex]
Therefore, the correct option from the given choices is:
[tex]\[
(z-5)(z-5)
\][/tex]
