Factoring x^5+x^4+1 but with only 0, 1, & 2

Overview

blackpenredpen explains how factoring over Z₃ restricts coefficients and calculations to the residues 0, 1, and 2, then applies synthetic division to x⁵ + x⁴ + 1. The polynomial has two roots at 1, giving the factorization (x + 2)²(x³ + 2x + 1) modulo 3; the remaining cubic has no root in Z₃.

Key takeaways

In Z₃, every integer coefficient is reduced to one of 0, 1, or 2, and every arithmetic operation is performed modulo 3.
The polynomial x⁵ + x⁴ + 1 has root 1 in Z₃ because substituting 1 gives 3 ≡ 0.
Synthetic division by 1 twice shows that (x − 1)² divides the polynomial over Z₃.
The remaining cubic is x³ + 2x + 1; testing the possible residues 0, 1, and 2 shows it has no root in Z₃.
Since −1 ≡ 2 modulo 3, the repeated factor can be written using the allowed residue representatives as (x + 2)².

Chapters

0:00 What Z₃ Means: Calculations with Residues 0, 1, and 2

Factoring over Z₃ requires polynomial coefficients and factors to use only the residues 0, 1, and 2.
Replace integers with their remainders after division by 3: for example, 17 ≡ 2 and 99 ≡ 0 modulo 3.
Negative values also reduce to residues: −1 ≡ 2 because adding 3 gives 2.
The congruence a ≡ b mod n means a = b + kn for some integer k.

2:40 Find the Root 1 and Divide x⁵ + x⁴ + 1

Since the constant term is 1, 0 cannot be a root; testing 1 gives 1 + 1 + 1 = 3 ≡ 0 modulo 3.
Synthetic division by the root 1 starts from coefficients 1, 1, 0, 0, 0, 1, with every sum reduced modulo 3.
The first division yields x⁴ + 2x³ + 2x² + 2x, so x − 1 is a factor in Z₃.

5:02 Extract the Repeated Root and State the Z₃ Factorization

Dividing the quartic by x − 1 again gives x³ + 2x + 1, establishing a repeated root at 1.
Testing 1 and 2 against the remaining cubic gives nonzero remainders, so neither residue is another root.
Because −1 ≡ 2 modulo 3, rewrite (x − 1)² as (x + 2)².
The resulting factorization is x⁵ + x⁴ + 1 ≡ (x + 2)²(x³ + 2x + 1) modulo 3.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, blackpenredpen.