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

Given:

Given expression is $3x^3 - x^2 - 3x + 1$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

Let $f(x)=3x^3 - x^2 - 3x + 1$

$f (1) = 3 (1)^3 - (1)^2 - 3 (1) + 1$

$= 3 - 1 - 3 + 1$

$= 0$

Therefore,  $(x - 1)$ is a factor of $f(x)$. 

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, 

$3x^3 - x^2 - 3x + 1 = (x - 1) (3x^2 + 2x - 1)$

$=(x-1)(3x^2 + 2x - 1)$

$= (x-1)(3x^2 + 3x - x - 1$

$=(x-1)[3x (x + 1) - 1 (x + 1)]$

$=(x-1) (3x - 1) (x + 1)$

Hence,  $3x^3 - x^2 - 3x + 1 = (x - 1) (x + 1) (3x - 1)$.

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Updated on: 10-Oct-2022

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