Question

How do you factor \[27{{x}^{3}}+8\]?

Answer 1

Hint: To solve the given question, we will need to use one of the algebraic expansion formulae. The expansion formula we will use is \[{{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\]. This expansion formula is known as the addition of cubes, as we are evaluating the addition of two perfect cubes.

Complete step by step answer:
The given expression is \[27{{x}^{3}}+8\], we have to factorize this expression. The given expression has two terms \[27{{x}^{3}}\], and 8. As we know that 27 is the cube of 3, hence we can express \[27{{x}^{3}}\] as \[{{3}^{3}}\times {{x}^{3}}\]. We know the property of exponents which states that, \[{{a}^{n}}{{b}^{n}}={{\left( ab \right)}^{n}}\]. Using this property, we can write \[27{{x}^{3}}\] as \[{{\left( 3x \right)}^{3}}\]. The second term of the expression is 8. As we know 8 is also a cube of 2, hence we can express 8 as \[{{\left( 2 \right)}^{3}}\].
Applying these two transformations for the given expression, we get
\[\Rightarrow 27{{x}^{3}}+8={{\left( 3x \right)}^{3}}+{{\left( 2 \right)}^{3}}\]
In this expression, two perfect cubes are being added, so this is an addition of cubes. This expression is of the form \[{{a}^{3}}+{{b}^{3}}\]. Here we have \[a=3x\], and \[b=2\]. We know the expansion for the addition of cubes is \[{{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\].

Substituting the values of a, and b for this expression, we get
\[\begin{align}
  & \Rightarrow 27{{x}^{3}}+8={{\left( 3x \right)}^{3}}+{{\left( 2 \right)}^{3}} \\
 & \Rightarrow \left( 3x+2 \right)\left( {{\left( 3x \right)}^{2}}-3x\times 2+{{2}^{2}} \right) \\
\end{align}\]
Simplifying the above expression, we get,
\[\Rightarrow \left( 3x+2 \right)\left( 9{{x}^{2}}-6x+4 \right)\]

Hence \[27{{x}^{3}}+8\] can be expressed in simplified form as \[\left( 3x+2 \right)\left( 9{{x}^{2}}-6x+4 \right)\].

Note: One should remember the expansion formula for addition of cubes, subtraction of cubes, subtraction of squares, etc. to solve these types of questions. There is a possibility of making calculation mistakes while finding squares, cube roots so it should be avoided.