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Factor completely: [tex]\(z^2 + 18z + 81\)[/tex]

Step 1: Determine whether the first term and the third term are perfect squares.
- The first term, [tex]\(z^2\)[/tex], is the square of [tex]\(z\)[/tex].
- The third term, [tex]\(81\)[/tex], is the square of [tex]\(9\)[/tex].

Step 2: Determine whether the middle term is 2 times the product of the expressions being squared in the first and last term.
- The middle term, [tex]\(18z\)[/tex], is 2 times the product of [tex]\(z\)[/tex] and [tex]\(9\)[/tex].

Step 3: Use [tex]\(A^2 + 2AB + B^2 = (A + B)^2\)[/tex] to factor the expression.
[tex]\[
z^2 + 18z + 81 = (z + 9)^2
\][/tex]

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Step-by-Step Solution:

Step 1: Identify the perfect squares.
- The first term, [tex]\( z^2 \)[/tex], is the square of [tex]\( z \)[/tex].
- The third term, 81, is the square of 9:

[tex]\[ z^2 = (z)^2 \][/tex]
[tex]\[ 81 = 9^2 \][/tex]

Step 2: Verify the middle term.
- The middle term, [tex]\( 18z \)[/tex], should be 2 times the product of [tex]\( z \)[/tex] and 9:

[tex]\[ 2 \cdot z \cdot 9 = 18z \][/tex]

Since [tex]\( 18z = 2 \cdot z \cdot 9 \)[/tex], we verify that the middle term is correctly formed.

Step 3: Use the perfect square trinomial formula.
- The quadratic expression fits the form [tex]\( A^2 + 2AB + B^2 = (A + B)^2 \)[/tex], where [tex]\( A = z \)[/tex] and [tex]\( B = 9 \)[/tex]:

[tex]\[ z^2 + 18z + 81 = (z + 9)^2 \][/tex]

So, the factored form of the given expression [tex]\( z^2 + 18z + 81 \)[/tex] is:

[tex]\[ (z + 9)^2 \][/tex]

Therefore, the completely factored expression is:

[tex]\[ z^2 + 18z + 81 = (z + 9)^2 \][/tex]
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