Let f(x)=x^3+x^2+x+1\in Z_2[x]. Write f(x) as a product of irreducible polynomials over Z_2

sanuluy

sanuluy

Answered question

2021-09-09

Let f(x)=x3+x2+x+1Z2[x]. Write f(x) as a product of irreducible polynomials over Z2

Answer & Explanation

Benedict

Benedict

Skilled2021-09-10Added 108 answers

Given:
f(x)=x3+x2+x+1
The given polynomial can be factorized as a product two irreducible polynomials as,
x3+x2+x+1=x2(x+1)+(x+1)
=(x+1)(x2+1)
Here, the polynomials (x+1) is in irreducible form which cannot be factorized further.
(x2+1) can be factorized over Z as,
x2+1=(x+i)(xi)
Here, both (x+i) and (x-i) are in irreducible form.
Hence, the given polynomial can be expressed as a product of irreducible polynomial over Z as,
x3+x2+x+1=(x+1)(xi)(x+i)

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