Factorizing a polynomial over a ring

[Adorno]

Homework Statement

Factorize $x^2 + x + 8$ in $\mathbb{Z}_{10}[x]$ in two different ways

Homework Equations

The Attempt at a Solution

I can see that x = 8 = -2 and x = 1 = -9 are roots of the polynomial, so one factorization is (x + 2)(x + 9).

Is there a systematic way to find all the factorizations?

 

[HallsofIvy]

I just looked at the various ways to get a product of 8 mod 10, the look at the sum of those products. I immediately see that 2*9= 18= 8 mod 10 and 2+ 9= 11= 1 mod 10 and that 4*7= 28= 8 mod 10 and that 4+ 7= 11= 1 mod 10.

 

[SammyS]

Homework Statement

Factorize $x^2 + x + 8$ in $\mathbb{Z}_{10}[x]$ in two different ways

Homework Equations

The Attempt at a Solution

I can see that x = 8 = -2 and x = 1 = -9 are roots of the polynomial, so one factorization is (x + 2)(x + 9).

Is there a systematic way to find all the factorizations?

It looks like the "Master Product rule"form introductory algebra --- only here you use modular arithmetic.

You are looking for a pair of numbers whose sum is 1 (mod 10) and whose product is 8 (mod 10).

 

[Adorno]

So, it's essentially just trial and error?

 

[SammyS]

So, it's essentially just trial and error?

Yes, but it's somewhat systematic trial & error.