Question

Factorise \[{a^3} + a - 3{a^2} - 3 \]

Answer 1

Hint: Factorization is a method of writing numbers as their factors or divisors. When the factors of a number are multiplied together, they give the original equation. In this question we can solve by grouping the \[{a^3} - 3{a^2} \] and \[a - 3 \] . After that we can take common and regrouping and we get the required solution.

Complete step-by-step answer:
Given, \[{a^3} + a - 3{a^2} - 3 \]
Grouping the terms \[{a^3} \] and \[ - 3{a^2} \] . Further grouping the terms \[a \] and \[ - 3 \] . We get,
 \[ = {a^3} - 3{a^2} + a - 3 \]
 \[ = ({a^3} - 3{a^2}) + (a - 3) \]
Taking \[{a^2} \] common in the term \[{a^3} - 3{a^2} \] . We get,
 \[ = {a^2}(a - 3) + a - 3 \]
As we can see that 1 is a multiple of any non-zero number. Using this we can take 1 as common in the term \[a - 3 \] .
 \[ = {a^2}(a - 3) + 1(a - 3) \]
Also we can see that \[a - 3 \] is present in both the terms, taking it common, we get:
 \[ = (a - 3)({a^2} + 1) \]
Hence factorization of \[{a^3} + a - 3{a^2} - 3 = (a - 3)({a^2} + 1) \]
Further if we want we can find the zeros or roots of the given equation by equating the factors to zero. Keeping variables on one side and constant on the other side we get the required answers.
So, the correct answer is “$(a - 3)({a^2} + 1)$”.

Note: In factorization we solve quadratic equations. The method of solving quadratic equations is to write the equation in the correct form. To be in the correct form, you must remove all parentheses from each side of the equation by distributing, combining all like terms and finally set the equation equal to zero. Use factoring strategies to factor the problem. Use the zero product property and set each factor containing a variable equal to zero. Solve each factor that was set equal to zero by getting the x on one side and the answer on the other side.