Question

Factorise the polynomial \[729{{x}^{3}}-512{{y}^{3}}\]

Answer 1

Hint: We solve this problem first by converting each term of the polynomial as the cube of some other term. Then we use the standard formula of algebra that is
\[{{a}^{3}}-{{b}^{3}}={{\left( a-b \right)}^{3}}+3ab\left( a-b \right)\]
After that, we take the common terms out if any to factorize the polynomial some more.

Complete step by step solution:
We are given that the polynomial as \[729{{x}^{3}}-512{{y}^{3}}\]
Let us assume that the given polynomial as
\[\Rightarrow p=729{{x}^{3}}-512{{y}^{3}}\]
Here, we can see that there are cubic terms of \[x,y\]
So, let us try to convert the given polynomial in such a way that each term can be written as cube of some other term
We know that
\[\begin{align}
  & \Rightarrow 729={{9}^{3}} \\
 & \Rightarrow 512={{8}^{3}} \\
\end{align}\]
By substituting the values in given polynomial we get
\[\Rightarrow p={{9}^{3}}\times {{x}^{3}}-{{8}^{3}}\times {{y}^{3}}\]
We know that the simple algebra formula that is
\[\Rightarrow {{a}^{n}}\times {{b}^{n}}={{\left( ab \right)}^{n}}\]
By using this formula in above equation we get
\[\Rightarrow p={{\left( 9x \right)}^{3}}-{{\left( 8y \right)}^{3}}\]
We know that the standard formula of algebra that is
\[{{a}^{3}}-{{b}^{3}}={{\left( a-b \right)}^{3}}+3ab\left( a-b \right)\]
By using this formula to given polynomial we get
\[\Rightarrow p={{\left( 9x-8y \right)}^{3}}+3\left( 9x \right)\left( 8y \right)\left( 9x-8y \right)\]
Now, by taking the common term out we get
\[\Rightarrow p=\left( 9x-8y \right)\left[ {{\left( 9x-8y \right)}^{2}}+3\left( 9x \right)\left( 8y \right) \right]\]
We know that the formula of square of difference of two numbers as
\[\Rightarrow {{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\]
By using this formula in above equation we get
\[\Rightarrow p=\left( 9x-8y \right)\left[ {{\left( 9x \right)}^{2}}-2\left( 9x \right)\left( 8y \right)+{{\left( 8y \right)}^{2}}+3\left( 9x \right)\left( 8y \right) \right]\]
By adding the required terms by multiplying the terms in above equation we get
\[\Rightarrow p=\left( 9x-8y \right)\left( 81{{x}^{2}}+72xy+64{{y}^{2}} \right)\]
Therefore the value of given polynomial after factorisation is given as
\[\therefore 729{{x}^{3}}-512{{y}^{3}}=\left( 9x-8y \right)\left( 81{{x}^{2}}+72xy+64{{y}^{2}} \right)\]

Note: We have a shortcut for solving this problem.
We have the given polynomial as
\[\Rightarrow p={{\left( 9x \right)}^{3}}-{{\left( 8y \right)}^{3}}\]
Now we have the direct formula for difference of cubes of two numbers as
\[\Rightarrow {{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]
By using the above formula directly we get
\[\Rightarrow p=\left( 9x-8y \right)\left[ {{\left( 9x \right)}^{2}}+\left( 9x \right)\left( 8y \right)+{{\left( 8y \right)}^{2}} \right]\]
By multiplying the required terms in above equation we get
\[\Rightarrow p=\left( 9x-8y \right)\left( 81{{x}^{2}}+72xy+64{{y}^{2}} \right)\]
Therefore the value of given polynomial after factorisation is given as
\[\therefore 729{{x}^{3}}-512{{y}^{3}}=\left( 9x-8y \right)\left( 81{{x}^{2}}+72xy+64{{y}^{2}} \right)\]