Question

Factorize the following expression as follows:
$27{{x}^{3}}-125{{y}^{3}}$
(a)$\left( 3x+5y \right)\left( 9{{x}^{2}}+25{{y}^{2}}+15xy \right)$
(b) $\left( 3x-5y \right)\left( 9{{x}^{2}}+25{{y}^{2}}-15xy \right)$
(c) $\left( 3x+5y \right)\left( 9{{x}^{2}}+25{{y}^{2}}-15xy \right)$
(d) $\left( 3x-5y \right)\left( 9{{x}^{2}}+25{{y}^{2}}+15xy \right)$

Answer 1

Hint: We have given an algebraic expression $27{{x}^{3}}-125{{y}^{3}}$ and we are asked to factorize it. If you look carefully then you will find that 27 can be written as ${{3}^{3}}$ and 125 as ${{5}^{3}}$. Then substituting these values in the given expression we get ${{\left( 3x \right)}^{3}}-{{\left( 5y \right)}^{3}}$ and then apply the identity in this expression which is equal to ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right)$. Now, substitute 3x in place of “a” and 5y in place of “b” in this identity and then you will get the required factorization.

Complete step-by-step answer:
The expression given in the above question is equal to:
$27{{x}^{3}}-125{{y}^{3}}$
As you can carefully look the above expression you will find that we can write 27 as ${{3}^{3}}$ and 125 as ${{5}^{3}}$ so rewriting the above expression we get,
$\begin{align}
  & {{3}^{3}}{{x}^{3}}-{{5}^{3}}{{y}^{3}} \\
 & ={{\left( 3x \right)}^{3}}-{{\left( 5y \right)}^{3}} \\
\end{align}$
The above expression is written in the form of ${{a}^{3}}-{{b}^{3}}$ which you can see that this form is the algebraic identity that is equal to:
${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right)$
Substituting “a” as 3x and “b” as 5y in the above equation we get,
$\begin{align}
  & {{\left( 3x \right)}^{3}}-{{\left( 5y \right)}^{3}}=\left( 3x-5y \right)\left( {{\left( 3x \right)}^{2}}+{{\left( 5y \right)}^{2}}+3x\left( 5y \right) \right) \\
 & \Rightarrow {{\left( 3x \right)}^{3}}-{{\left( 5y \right)}^{3}}=\left( 3x-5y \right)\left( 9{{x}^{2}}+25{{y}^{2}}+15xy \right) \\
\end{align}$
From the above solution, we have factorized the given expression as $\left( 3x-5y \right)\left( 9{{x}^{2}}+25{{y}^{2}}+15xy \right)$. Now, comparing this result with the options given in the above problem we find that option (d) is correct.
Hence, option (d) is the correct option.

Note: The question demands the knowledge of the algebraic identities like in this problem we have used the algebraic identity ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right)$ if you don’t know this identity then it will be very difficult for you to solve this question.
The other similar algebraic identity that you can use in some other questions is:
${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}+{{b}^{2}}-ab \right)$
For e.g. in some other question you are given the following expression and you are asked to factorize it:
$27{{x}^{3}}+125{{y}^{3}}$
Rewriting the above expression we get,
${{\left( 3x \right)}^{3}}+{{\left( 5x \right)}^{3}}$
Now, using the new algebraic identity that we have shown above we can write the above expression as:
$\begin{align}