Question

order the steps to factor x^4 - 1 completely using polynomial identities from top to bottom

Answer 1

To factor \( x^4 - 1 \) completely using polynomial identities, you can follow these steps:

1. **Recognize that \( x^4 - 1 \) is a difference of squares:**
\[
x^4 - 1 = (x^2)^2 - 1^2
\]
This allows you to apply the difference of squares identity \( a^2 - b^2 = (a - b)(a + b) \).

2. **Apply the difference of squares:**
\[
x^4 - 1 = (x^2 - 1)(x^2 + 1)
\]

3. **Factor \( x^2 - 1 \) further, as it is also a difference of squares:**
\[
x^2 - 1 = (x - 1)(x + 1)
\]

4. **Combine all the factors:**
The polynomial \( x^4 - 1 \) can now be written as:
\[
x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)
\]

5. **Check if \( x^2 + 1 \) can be factored further:**
\( x^2 + 1 \) is irreducible over the real numbers, since it does not have real roots. However, if factoring over the complex numbers, it can be written as:
\[
x^2 + 1 = (x - i)(x + i)
\]

6. **Final factored form (over the complex numbers):**
If you include complex factors, you can express the complete factorization as:
\[
x^4 - 1 = (x - 1)(x + 1)(x - i)(x + i)
\]

**Summary of the steps in order:**
1. Identify as a difference of squares.
2. Factor using the difference of squares identity.
3. Factor \( x^2 - 1 \) again as a difference of squares.
4. Combine factors.
5. (Optional) Factor \( x^2 + 1 \) over the complex numbers if desired.