Given polynomial x^4+4x^3+4x^2+1 The task is find expansion of the polynomial as a product of irreducible polynomials in RR

John Landry 2022-07-22 Answered
Given polynomial
x 4 + 4 x 3 + 4 x 2 + 1.
The task is find expansion of the polynomial as a product of irreducible polynomials in R
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Answers (1)

Helena Howard
Answered 2022-07-23 Author has 12 answers
It suffices to show that f ( x ) = x 4 + 4 x 3 + 4 x 2 + 1 has no linear factors over Z 3 :
f ( 0 ) = f ( 1 ) = 1 and f ( 2 ) = 2, so f ( x ) has no linear factors. Then f ( x ) must factor to two quadratic polynomials:
f ( x ) = ( a x 2 + b x + c ) ( u x 2 + v x + w )
We then have that a u = 1. Multiplying the first polynomial by u and the second by a, we may assume that a = u = 1. Equating the coefficients of the powers of x, we have
4 = v + b 4 = w + c + b v 0 = b w + c v 1 = c w
Some algebra shows that
b = 2 + 2 ( 1 + 2 ) c = 1 + 2 + 2 ( 1 + 2 ) v = 2 2 ( 1 + 2 ) w = 1 + 2 2 ( 1 + 2 )
So, letting α = 1 + 2 , we see that x 4 + 4 x 3 + 4 x 2 + 1 factors to
( x 2 + ( 2 + 2 α ) x + α + 2 α ) ( x 2 + ( 2 2 α ) x + α 2 α ) .
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