Factorise: `x^(3)-3x^(2) -x +3`

Answer

To factorise the expression \( x^3 - 3x^2 - x + 3 \), we will follow these steps:

### Step 1: Group the terms
We start by grouping the terms in pairs:
\[ (x^3 - 3x^2) + (-x + 3) \]

### Step 2: Factor out common terms from each group
From the first group \( x^3 - 3x^2 \), we can factor out \( x^2 \):
\[ x^2(x - 3) \]
From the second group \( -x + 3 \), we can factor out \( -1 \):
\[ -1(x - 3) \]

Now, we rewrite the expression:
\[ x^2(x - 3) - 1(x - 3) \]

### Step 3: Factor out the common binomial factor
Now we see that \( (x - 3) \) is a common factor:
\[ (x - 3)(x^2 - 1) \]

### Step 4: Recognize and apply the difference of squares
The expression \( x^2 - 1 \) can be factored further using the difference of squares identity \( a^2 - b^2 = (a - b)(a + b) \):
\[ x^2 - 1 = (x - 1)(x + 1) \]

### Step 5: Combine the factors
Now we can write the complete factorization:
\[ (x - 3)(x - 1)(x + 1) \]

Thus, the final factorization of the expression \( x^3 - 3x^2 - x + 3 \) is:
\[ (x - 3)(x - 1)(x + 1) \]

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