Question

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

Answer 1

Hint: A polynomial can only be factorized if it has real roots. For example the quadratic expression \[{{x}^{2}}+1\] can not be further factorized, as it has no real roots. The algebraic expression of the form \[{{a}^{2}}-{{b}^{2}}\]. The factored form of these types of expression is \[\left( a+b \right)\left( a-b \right)\]. The factored form of a polynomial expression is used to find the roots of the polynomial.

Complete step by step solution:
The given expression is \[{{n}^{4}}-1\]. It has two terms; the first term is \[{{n}^{4}}\] and the second term is 1.
As, we know that 1 is square of itself, the first term can also be written as \[{{\left( {{n}^{2}} \right)}^{2}}\]. Using this simplification in the given expression, it can be written as \[{{n}^{4}}-1={{\left( {{n}^{2}} \right)}^{2}}-{{1}^{2}}\].
As we can see that this expression is evaluating the difference of two square terms, it is a difference of square form. We know that the difference of square expression \[{{a}^{2}}-{{b}^{2}}\] is factorized as \[\left( a+b \right)\left( a-b \right)\]. Here, we have a, and b are \[{{n}^{2}}\] and 1 respectively. Substituting the values in the expansion, we get
\[{{n}^{4}}-1=\left( {{n}^{2}}+1 \right)\left( {{n}^{2}}-1 \right)\]
Here the first factor term is \[{{n}^{2}}+1\]. As, we know that it can not be further factorized, as it has no real roots. The second factor term is \[{{n}^{2}}-1\]. This quadratic equation has two real roots: they are \[1\And -1\]. So, it can be expressed as \[\left( n+1 \right)\left( n-1 \right)\].
Hence the factored form of the given expression is \[\left( {{n}^{2}}+1 \right)\left( n+1 \right)\left( n-1 \right)\].

Note: The maximum number of factors a polynomial can have is equals to its degree. This will happen only if all the roots of the polynomial are real. In this example, we get three factors but the degree of this expression is four.