SOLUTION: The polynomial in Q[t] determine the rational roots(if any) and factor the polynomial as a product of irreducible polynomials in Q[t]. f(t)= t^5 - 2t^4 - 4t^3 + 4t^2

SOLUTION: The polynomial in Q[t] determine the rational roots(if any) and factor the polynomial as a product of irreducible polynomials in Q[t]. f(t)= t^5 - 2t^4 - 4t^3 + 4t^2 - 5t +6

Question 31284: The polynomial in Q[t] determine the rational roots(if any) and factor the polynomial as a product of irreducible polynomials in Q[t].
f(t)= t^5 - 2t^4 - 4t^3 + 4t^2 - 5t +6
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
f(t)= t^5 - 2t^4 - 4t^3 + 4t^2 - 5t +6
we find f(1)=1-2-4+4-5+6=0..so x-1 is a factor
1...|1.......-2......-4.......4.........-5.........6
....|0.......1.......-1......-5.........-1........-6
----------------------------------------------------------
-2..|1.......-1......-5......-1.........-6.........0..WE FIND F(-2)=0.
....|0.......-2......6........-2........6......
---------------------------------------------------------
3...|1.......-3.......1.......-3........0...........WE FIND F(3)=0
....|0.......3........0........3.............
-----------------------------------------------------
....|1.......0.........1.......0..............................
T^2+1 IS QUOTIENT.THIS IS NOT REDUCIBLE OVER REAL NUMBERS ...HENCE
F(T)=(T-1)(T+2)(T-3)(T^2+1)
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