Question

Factorize $8{x^3} + 36{x^2}y + 54x{y^2} + 27{y^3}$.

Answer 1

Hint: Here, we are given a polynomial and we need to factorize it. Factors means which when divided by the given polynomial gives remainder as 0. To factorize $8{x^3} + 36{x^2}y + 54x{y^2} + 27{y^3}$, we will make some adjustments in the polynomial.

Formula used:
$ {\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$

Complete step by step solution:
In this question, we are given a polynomial and we need to factorize it.
The given polynomial is : $8{x^3} + 36{x^2}y + 54x{y^2} + 27{y^3}$ - - - - - - - - - - - - - - (1)
Factors of a polynomial means which when divided by the given polynomial leaves remainder 0.
Here, to factorize the given polynomial $8{x^3} + 36{x^2}y + 54x{y^2} + 27{y^3}$, we are going to use the formula of ${\left( {a + b} \right)^3}$. For that we need to make some adjustments.
Here, we can write $8{x^3}$ as ${\left( {2x} \right)^3}$ and we can write $36{x^2}y$ as $3 \times {\left( {2x} \right)^2} \times 3y$ and $54x{y^2}$ as $3 \times 2x \times {\left( {3y} \right)^2}$ and $27{y^3}$ as ${\left( {3y} \right)^3}$. Therefore, substituting these values in equation (1) , the equation becomes
$ \Rightarrow 8{x^3} + 36{x^2}y + 54x{y^2} + 27{y^3} = {\left( {2x} \right)^3} + {\left( {3y} \right)^3} + 3 \times {\left( {2x} \right)^2} \times 3y + 3 \times \left( {2x} \right) \times {\left( {3y} \right)^2}$- - - - - - - - - - - - - (2)
Now, we know the formula of ${\left( {a + b} \right)^3}$ is
$ \Rightarrow {\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$- - - - - - - - - - - - - - - (3)
Here, we have $a = 2x$ and $b = 3y$
Therefore, on comparing the equations (2) and (3), we get
\[ \Rightarrow {\left( {2x} \right)^3} + {\left( {3y} \right)^3} + 3 \times {\left( {2x} \right)^2} \times 3y + 3 \times \left( {2x} \right) \times {\left( {3y} \right)^2} = {\left( {2x + 3y} \right)^3}\]
Hence, we have factorized
\[ \Rightarrow 8{x^3} + 36{x^2}y + 54x{y^2} + 27{y^3} = {\left( {2x + 3y} \right)^3}\]

Therefore, the factors of $8{x^3} + 36{x^2}y + 54x{y^2} + 27{y^3}$ are $\left( {2x + 3y} \right)\left( {2x + 3y} \right)\left( {2x + 3y} \right)$.

Note:
Generally, polynomials can be factorized using the Polynomial remainder theorem.
Let us see what this theorem is.
For a given polynomial, $x - a$ is a divisor of f(x) if $f\left( a \right) = 0$. That means if we are given a polynomial and if we put a in place of x in the given polynomial and if we get value 0, then $x - a$ will be one of the factors of the given polynomial.