Question

Factorize: $27{x^3} - 125{y^3}$
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:
Here, we have to find the factors of the given algebraic expression. We will first simplify the given expression by writing them as a cube of some number. Then we will factorize it by using the suitable algebraic identity to get the required answer. An algebraic expression is an expression with the combination of variables, constants and operators. Factorization is a process of rewriting the expression in terms of the product of the factors.

Formula Used:
The difference between the cube of two numbers is given by the algebraic identity ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$ where $a,b$ be the two numbers respectively.

Complete step by step solution:
We are given an algebraic expression $27{x^3} - 125{y^3}$.
We know that $27{x^3}$ can be represented as ${\left( {3x} \right)^3}$, whereas $125{y^3}$ can be represented as ${\left( {5y} \right)^3}$.
Thus, we get
$27{x^3} - 125{y^3} = {\left( {3x} \right)^3} - {\left( {5y} \right)^3}$
The given algebraic expression $27{x^3} - 125{y^3}$is of the form ${a^3} - {b^3}$.
Now, by using the algebraic identity ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$, we get
$ \Rightarrow 27{x^3} - 125{y^3} = \left( {3x - 5y} \right)\left( {{{\left( {3x} \right)}^2} + \left( {3x} \right)\left( {5y} \right) + {{\left( {5y} \right)}^2}} \right)$
Applying the exponent on the terms and multiplying the terms, we get
$ \Rightarrow 27{x^3} - 125{y^3} = \left( {3x - 5y} \right)\left( {9{x^2} + 15xy + 25{y^2}} \right)$
Therefore, the factors of $27{x^3} - 125{y^3}$ is $\left( {3x - 5y} \right)\left( {9{x^2} + 15xy + 25{y^2}} \right)$.

Thus, option (D) is the correct answer.

Note:
We know that factors are numbers if the expression is a numeral. Factors are algebraic expressions if the expression is an algebraic expression. Factorization is done by using the common factors, the grouping of terms and the algebraic identity. We know that an equality relation which is true for all the values of the variables is called an Identity. We should be careful that the algebraic expression has to be rewritten in the form of algebraic identity. If any factor is common, then it can be taken out as a common factor and check whether all the terms after taking the common factor are in the form of Algebraic Identity.