Factorise the following :
`x^(2)-16x+39`

Answer

To factorise the expression \( x^2 - 16x + 39 \), we will use the method of splitting the middle term. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -16 \) (coefficient of \( x \))
- \( c = 39 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( a \cdot c = 1 \cdot 39 = 39 \) and add up to \( b = -16 \).

The two numbers that satisfy these conditions are \( -13 \) and \( -3 \) because:
- \( -13 \times -3 = 39 \)
- \( -13 + (-3) = -16 \)

### Step 3: Rewrite the middle term
We can rewrite the expression by splitting the middle term using the two numbers we found:
\[ x^2 - 13x - 3x + 39 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (x^2 - 13x) + (-3x + 39) \]

### Step 5: Factor out the common terms from each group
From the first group \( (x^2 - 13x) \), we can factor out \( x \):
\[ x(x - 13) \]
From the second group \( (-3x + 39) \), we can factor out \( -3 \):
\[ -3(x - 13) \]

### Step 6: Combine the factored terms
Now we can combine the two groups:
\[ x(x - 13) - 3(x - 13) \]
We can factor out the common term \( (x - 13) \):
\[ (x - 13)(x - 3) \]

### Final Answer
Thus, the factorised form of the expression \( x^2 - 16x + 39 \) is:
\[ (x - 13)(x - 3) \]

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