Let, f (x) = 3x³-x²-3x+1

The factors of the constant term

The factor of the coefficient of x³ is 3. Hence, possible rational roots of f (x) are:

We have,

f (1) = 3 (1)³ – (1)² – 3 (1) + 1

= 3 – 1 – 3 + 1

= 0

So, (x – 1) is a factor of f (x)

Let us now divide

f (x) = 3x³-x²-3x+1 by (x - 1) to get the other factors of f (x)

Using long division method, we get

3x³-x²-3x+1 = (x – 1) (3x² + 2x – 1)

Now,

3x² + 2x - 1 = 3x² + 3x – x – 1

= 3x (x + 1) – 1 (x + 1)

= (3x – 1) (x + 1)

Hence, 3x³-x²-3x+1 = (x – 1) (x + 1) (3x – 1)