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

Expert Answer to "Let f(x)=x3+x2+x+1∈Z2[x]. Write f(x) as a product of..."

sanuluy

2021-09-09

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}

Benedict

Skilled2021-09-10Added 108 answers

Given:

f (x) = x^{3} + x^{2} + x + 1

The given polynomial can be factorized as a product two irreducible polynomials as,

x^{3} + x^{2} + x + 1 = x^{2} (x + 1) + (x + 1)

= (x + 1) (x^{2} + 1)

Here, the polynomials (x+1) is in irreducible form which cannot be factorized further.

(x^{2} + 1)

can be factorized over Z as,

x^{2} + 1 = (x + i) (x - i)

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,

x^{3} + x^{2} + x + 1 = (x + 1) (x - i) (x + i)

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