Question

How can you factorize the expression \[27{x^2} - 3\] ?

Answer 1

Hint:This is a problem of factorization of a given expression. At first we need to find the value which is the common factor of both the terms. After taking that value as a common term we can further reduce the remaining using algebraic identity. We need to factorize it to the simplest possible factors.

Formula used: In this given problem we have used only the algebraic identity for the difference of two square terms which is as follows. \[({a^2} - {b^2}) = (a + b)(a - b)\]

Step by step solution:
Firstly, we need to find the common factor of both terms. In \[27{x^2} - 3\]
we see that \[3\] is the common term among \[27{x^2}\,and\,3.\]
So, we take \[3\] as a common factor from both the terms.

Hence the expression reduces to:
 \[27{x^2} - 3 = 3(9{x^2} - 1)\]

Now the term \[(9{x^2} - 1)\] can be expressed as the difference of two squares as: \[\left[
{{{(3x)}^2} - {{(1)}^2}} \right]\] .

We know the algebraic identity for the difference of two squares. It is expressed as: \[({a^2} - {b^2}) = (a + b)(a - b)\]

According to our question, \[a = 3x\,and\,b = 1\] . So we can further factorise this term into two simple factors.

Thus , \[\left[ {{{(3x)}^2} - {{(1)}^2}} \right]\] \[ = (3x + 1)(3x - 1)\]

Therefore, the overall expression can be expressed as:
 \[27{x^2} - 3 = 3(3x + 1)(3x - 1)\]

The entire expression is broken into simplest 3 factors. It is not possible to further factorize into simplest terms by any algebraic means. Hence this is the final result.

Hence, the simple factored expression for the term \[27{x^2} - 3\] is \[3(3x + 1)(3x - 1)\] .

Note:Any expression of the type \[({a^2} - {b^2})\] can be factored as \[(a + b)(a - b)\] . The terms
 \[a\,and\,b\] may not have rational square roots. In that case we can express the factors as their irrational square roots. Many other identities may also be used to factorize the given expression.

There are also algebraic identities for the difference for cubes of two terms. We should try to reduce the expression to an extent such that the variables possess the least whole value.