Question

How do you factor \[{x^3} - 1\]?

Answer 1

Hint: In the given question, we have been given a polynomial. The degree of the polynomial is three. We have to factorize the given polynomial. The first part of the polynomial (\[{x^3}\]) which is a cube, is a cube of the variable \[x\]. While the second part of the polynomial (1) is a natural number. And, this number is also a cube, it is the cube of 1. The two parts of the polynomial are separated by a basic arithmetic operator – the minus “–” sign. So, we can easily factorize the given polynomial by applying the formula of difference of two cubes.

Formula Used:
In the given question, we are going to apply the formula of difference of two cubes.
Let there be two integral numbers; say the numbers are \[a\] and \[b\], the difference of their cubes can be written as:
\[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\]

Complete step-by-step answer:
In the given question, we are going to apply the formula of difference of two cubes.
Let there be two integral numbers; say the numbers are \[a\] and \[b\].
Now, the difference of their cubes can be represented as,
\[{a^3} - {b^3}\]
Now, there is a standard formula for the factorization of the two cubes, which is:
\[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\]
So, to solve this question, we just put \[a = x\] and \[b = 1\], and we get:
\[{x^3} - 1 = \left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\]
Hence, we can factor \[{x^3} - 1\] as \[\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\].

Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. In the given question, we were given a polynomial. We first found what type of polynomial it is. Then we wrote down the formula which can be used to solve, or factorize the given polynomial. Then we just put in the values from the question, and just solve it like normal.