Factorise:
`z ^(2) - 12z- 45`

Answer

To factorise the expression \( z^2 - 12z - 45 \), we will follow these steps:

### Step 1: Identify the coefficients
The given expression is \( z^2 - 12z - 45 \). Here, the coefficient of \( z^2 \) is 1, the coefficient of \( z \) is -12, and the constant term is -45.

### Step 2: Multiply the coefficient of \( z^2 \) by the constant term
We multiply the coefficient of \( z^2 \) (which is 1) by the constant term (-45):
\[ 1 \times (-45) = -45 \]

### Step 3: Find two numbers that multiply to -45 and add to -12
We need to find two numbers that multiply to -45 and add up to -12. The two numbers that satisfy these conditions are -15 and 3:
\[ -15 \times 3 = -45 \quad \text{and} \quad -15 + 3 = -12 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the expression \( z^2 - 12z - 45 \) as:
\[ z^2 - 15z + 3z - 45 \]

### Step 5: Factor by grouping
Now, we will group the terms:
\[ (z^2 - 15z) + (3z - 45) \]
Next, we factor out the common factors from each group:
\[ z(z - 15) + 3(z - 15) \]

### Step 6: Factor out the common binomial
Now, we can factor out the common binomial \( (z - 15) \):
\[ (z - 15)(z + 3) \]

### Final Result
Thus, the factorised form of \( z^2 - 12z - 45 \) is:
\[ (z - 15)(z + 3) \]

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