Question

How do you factor \[{x^4} - 1\]?

Answer 1

Hint: In the given question, we have been asked to factorize a polynomial consisting of a variable and a constant. The polynomial is clearly of degree four. The first term is a variable raised to fourth power, but if we see closely, the second term (constant) is also a number raised to fourth power. So, we have to apply the formula of difference of two squares, twice.

Formula Used:
We are to apply the formula of difference of two squares:
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]

Complete step-by-step answer:
The polynomial to be factored is \[{x^4} - 1 = {x^4} - {1^4}\].
Clearly, \[{x^4} - {1^4} = {\left( {{x^2}} \right)^2} - {\left( {{1^2}} \right)^2}\]
So, we apply the formula of difference of two squares twice,
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
Hence, \[{x^4} - 1 = \left( {{x^2} + 1} \right)\left( {{x^2} - 1} \right)\]
Now, \[{x^2} - 1 = \left( {x + 1} \right)\left( {x - 1} \right)\]
Thus, \[{x^4} - 1 = \left( {{x^2} + 1} \right)\left( {x + 1} \right)\left( {x - 1} \right)\]

Additional Information:
We only have the simplified formula of difference of two squares, \[{a^2} - {b^2}\]. We do not have the formula for the sum of two squares, \[{a^2} + {b^2}\]. For that we have a combination of two formulae – \[{a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab\].

Note: When the number is raised to some power whose formula is not known, we try to break it down to the ones whose formula is known and solve it accordingly.