Question

# Which of the following is the correct complete factorization of x^{4} - 1?

Polynomial factorization
Which of the following is the correct complete factorization of $$x^{4} - 1$$?

2021-05-26
Step 1
To find the complete factorization of $$x^{4}−1$$.
Solution:
We know that,
$$a^{2}-b^{2}=(a=b)(a-b)$$
Factorizing the given expression using above identity.
$$x^{4}-1=(x^{2})^{2}-1^{2}$$
$$=(x^{2}+1)(x^{2}-1)$$
$$=(x^{2}+1)(x^{2}-1^{2})$$
$$=(x^{2}+1)(x+1)(x-1)$$
Therefore, $$x^{4}-1=(x^{2}+1)(x+1)(x-1)$$.
Step 2
Hence, complete factorization of $$x^{4}-1\ is\ (x^{2}+1)(x+1)(x-1)$$.