Factorise:
`x ^(2) - 23x + 42`

Answer

To factorise the expression \( x^2 - 23x + 42 \), we will follow these steps:

### Step 1: Identify the coefficients
The given expression is \( x^2 - 23x + 42 \). Here, the coefficient of \( x^2 \) is 1, the coefficient of \( x \) is -23, and the constant term is 42.

### Step 2: Find two numbers that multiply to the constant term and add to the coefficient of \( x \)
We need to find two numbers that multiply to 42 (the constant term) and add up to -23 (the coefficient of \( x \)).

The pairs of factors of 42 are:
- \( 1 \times 42 \)
- \( 2 \times 21 \)
- \( 3 \times 14 \)
- \( 6 \times 7 \)

From these pairs, we see that \( -21 \) and \( -2 \) multiply to give \( 42 \) and add to give \( -23 \):
\[ -21 + (-2) = -23 \]

### Step 3: Rewrite the middle term using the two numbers found
We can rewrite the expression \( x^2 - 23x + 42 \) as:
\[ x^2 - 21x - 2x + 42 \]

### Step 4: Factor by grouping
Now, we will group the terms:
\[ (x^2 - 21x) + (-2x + 42) \]

Next, we factor out the common factors in each group:
\[ x(x - 21) - 2(x - 21) \]

### Step 5: Factor out the common binomial factor
Now, we can see that \( (x - 21) \) is a common factor:
\[ (x - 21)(x - 2) \]

### Final Answer
Thus, the factorised form of \( x^2 - 23x + 42 \) is:
\[ (x - 21)(x - 2) \]

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