Factorise:
`z ^(2) + 12z+27`

Answer

To factorise the expression \( z^2 + 12z + 27 \), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the standard quadratic form \( az^2 + bz + c \), where:
- \( a = 1 \) (coefficient of \( z^2 \))
- \( b = 12 \) (coefficient of \( z \))
- \( c = 27 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times 27 = 27 \) and add to \( b = 12 \).

The pairs of factors of 27 are:
- \( 1 \times 27 \)
- \( 3 \times 9 \)

Among these, \( 3 \) and \( 9 \) add up to \( 12 \):
- \( 3 + 9 = 12 \)

### Step 3: Rewrite the middle term
Now, we can rewrite the expression \( z^2 + 12z + 27 \) using the numbers we found:
\[ z^2 + 3z + 9z + 27 \]

### Step 4: Factor by grouping
Next, we will group the terms:
\[ (z^2 + 3z) + (9z + 27) \]

Now, we factor out the common factors in each group:
- From \( z^2 + 3z \), we can factor out \( z \):
\[ z(z + 3) \]
- From \( 9z + 27 \), we can factor out \( 9 \):
\[ 9(z + 3) \]

### Step 5: Combine the factors
Now we can combine the factored groups:
\[ z(z + 3) + 9(z + 3) \]
This can be factored further as:
\[ (z + 3)(z + 9) \]

### Final Answer
Thus, the factorised form of \( z^2 + 12z + 27 \) is:
\[ (z + 3)(z + 9) \]

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