Question

Factorise the cubic polynomial: $3{x^3} - {x^2} - 3x + 1$

Answer 1

Hint: In the given question, we have to factorise the given algebraic expression. First, we will take the common terms outside the bracket and simplify the expression. Then, we will apply the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ so as to factorise the expression further.

Complete step-by-step solution:
Given question requires us to factorize the expression: $3{x^3} - {x^2} - 3x + 1$.
So, let the expression $3{x^3} - {x^2} - 3x + 1$ be $F$.
So, we will take the common terms outside the bracket so as to factorize the expression and do simplification. Then, we will get,
$F = 3{x^3} - {x^2} - 3x + 1$
Taking ${x^2}$ from the first two terms and negative sign from last two terms, we get,
$ \Rightarrow F = {x^2}\left( {3x - 1} \right) - \left( {3x - 1} \right)$
Taking the expression $\left( {3x - 1} \right)$ common from the brackets, we get,
\[ \Rightarrow F = \left( {3x - 1} \right)\left( {{x^2} - 1} \right)\]
Now, we can factorise the given expression using the algebraic identity \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\]. So, we condense the entire algebraic expression using the identity to get,
\[ \Rightarrow F = \left( {3x - 1} \right)\left( {x - 1} \right)\left( {x + 1} \right)\]
So, the factored form of the given expression $F = 3{x^3} - {x^2} - 3x + 1$ is \[\left( {3x - 1} \right)\left( {x - 1} \right)\left( {x + 1} \right)\].
This is our required answer.

Note: We must know the applications of the algebraic identities so as to solve such problems. One must take care of the calculations while taking common terms out of the brackets and factoring the expression so as to be sure of the answer. We must know the simplification rules to obtain the final answer.