Factorise: \[{x^3} - 3{x^2} + x - 3\]
A) \[\left( {{x^2} + 7} \right)\left( {x - 3} \right)\]
B) \[\left( {{x^2} + 1} \right)\left( {x - 3} \right)\]
C) \[\left( {{x^2} - 1} \right)\left( {x - 2} \right)\]
D) \[\left( {{x^2} + 7} \right)\left( {x - 2} \right)\]

Hint: Here, we have to factorize the equation. We will be factoring the equation by factoring of common terms. Factorization or factoring is defined as the breaking or decomposition of an entity which may be a number, a matrix, or a polynomial into a product of another entity, or factors, which when multiplied together give the original number or a matrix.

Complete step by step solution:
We are given an equation \[{x^3} - 3{x^2} + x - 3\].
Now factoring of common terms, we get
\[ \Rightarrow {x^3} - 3{x^2} + x - 3 = 0\]
\[ \Rightarrow {x^2}(x - 3) + 1\left( {x - 3} \right) = 0\]
\[ \Rightarrow \left( {{x^2} + 1} \right)\left( {x - 3} \right) = 0\]

Therefore, the factorization of the equation \[{x^3} - 3{x^2} + x - 3\] is \[\left( {{x^2} + 1} \right)\left( {x - 3} \right)\].

Additional Information:
Factoring out common factors is applicable only If each term in the polynomial shares a common factor. The sum-product pattern is applicable only If the polynomial is of the form \[{x^2} + bx + c\], and there are factors of \[c\] that add up to \[b\]. The grouping method is applicable only If the polynomial is of the form \[a{x^2} + bx + c\], and there are factors of \[ac\] that add up to \[b\]. Perfect square trinomials are applicable only If the first and last terms are perfect squares and the middle term is twice the product of their square roots. Difference of squares is applicable only If the expression represents a difference of squares.

Note:
Factoring Polynomials by Grouping method is also said to be factoring by pairs. Here, the given polynomial is distributed in pairs or grouped in pairs to find the zeros. The factorization can be done also by using algebraic identities. We should know the importance of factoring. Factoring is an important process that helps us understand more about our equations. Through factoring, we rewrite our polynomials in a simpler form, and when we apply the principles of factoring to equations, we yield a lot of useful information.