Using factor theorem, factorize each of the following polynomials : `3x^3-x^2-3x+1`

Explanation

The factor of the coefficient of `x^3` is 3.
Hence, possible rational roots of f (x) are:`±1 ,±13 ±1,±13`
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) = 3x^3 - x^2 - 3x + 1 by (x - 1)` to get the other factors of f (x) Using long division method, we get `3x^3 - x^2 - 3x + 1`
`=> (x – 1) (3x^ 2 + 2x – 1) `
`=> Now, 3x^2 + 2x - 1`
`=> 3x^2 + 3x – x – 1`
`=> 3x (x + 1) – 1 (x + 1)`
`=> (3x – 1) (x + 1) `
`=> Hence, 3x^3 - x^2 - 3x + 1 = (x – 1) (x + 1) (3x – 1) `.