Question

How do you factor \[{x^3} + 15{x^2} + 75x + 125\]?

Answer 1

Hint: In the given question, we have been given a polynomial. It is a polynomial of the third degree. We have to factorize the polynomial. There is no direct formula for it. We are going to arrange the terms in a way that the two given cubes are together and the other two are together. Then we are going to factorize the cubes using the appropriate formula and factorize the other two terms by taking common factors. Then we are going to just simplify by taking the common terms from their separate factorization.

Formula Used:
We are going to use the sum of cubes formula:
\[{x^3} + {y^3} = \left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)\]

Complete step by step answer:
The given equation is \[{x^3} + 15{x^2} + 75x + 125\].
Let us re-arrange the terms:
\[{x^3} + 125 + 15{x^2} + 75x\]
We know, \[{5^3} = 125\], so
\[{x^3} + {5^3} + 15x\left( {x + 5} \right)\]
Applying the sum of cubes formula,
\[\left( {x + 5} \right)\left( {{x^2} - 5x + 25} \right) + \left( {x + 5} \right)15x\]
Taking common from the two terms,
\[\left( {x + 5} \right)\left( {{x^2} + 10x + 25} \right)\]
Now, factoring the second bracket by splitting the middle term,
\[{x^2} + 5x + 5x + 25 = {\left( {x + 5} \right)^2}\]

Hence, \[{x^3} + 15{x^2} + 75x + 125 = {\left( {x + 5} \right)^3}\]

Note:
In the given question, we had to factorize a cubic polynomial. To do that, we took the cubic terms together, factorized them, factorized the other two terms, took the common terms out and simplified the obtained expression. So, it is really important that we know the formulae and where, when and how to use them so that we can get the correct result.