# Thread: Show polynomials in Zn[x] as products of linear factorizations

1. ## Show polynomials in Zn[x] as products of linear factorizations

I have two questions on this topic. The first one is finding the x3+2 as a product of linear factors in Z3[X]. I have in my notes that -

x3+2=x3-1 (in mod 3)
=(x)3-(1)3
=(x3-1)(x2+x+1)
=(x-1)(x2+x+1)
=(x-1)(x-1)(x-1)

The part of this I dont get is the second and third step. How do we go from x3-1 to (x)3-(1)3=(x3-1)(x2+x+1)?

The second question is to show 2x3+3x-7x-5 as a product of linear factors in Z1[x]. I keep running into polynomials that cant be factored. I presume the step Im missing is just like the previous question where i need to use the properties of mod, but Im not sure how/where to do it?

2. ## Re: Show polynomials in Zn[x] as products of linear factorizations

Hi jquinc21,

The reason you're having trouble seeing the second step is because it's not true. If you multiply out $(x^{3} -1)(x^{2}+x+1)$ you will have a polynomial of degree 5 in $\mathbb{Z}_{3}[x],$ which is not equal to the degree 3 polynomial $x^{3}-1.$

Does this clear things up? Let me know if there is a point of confusion.

Also, not sure if you have seen this or not, but it's worth checking out the "Freshmen's Dream" theorem Freshman's dream - Wikipedia, the free encyclopedia. Since 3 is a prime this theorem tells us that IN $\mathbb{Z}_{3}[x]$, $(x-1)^{3}=x^{3}-1,$ which was what you concluded in your post above.