Question

Descompuneti in factori ireductibili polinomul:
a)X^3 2X (X-2) - 8=
b) X^6-4X^3+4=
c) X^5-X^3+5X2+5X=
d) X^4 -X^2 +2X+2=

Answer 1

x³+2x(x-2)-8= x³-8 +2x(x-2)=(x-2) (x²+2x+4)+2x(x-2)=(x-2)(x²+2x+4+2x)=(x-2)(x²+4x+4)=(x-2)(x+2)²
ireductibil in R



b)
X^6-4X^3+4=(x³-2)² ireductibil in Q
 in R
(x-∛2)²(x²+∛2 x+∛4)² ireductibil in R


c)
X^5-X^3+5X²+5X=x³(x²-1)+5x(x+1)=x³(x+1)(x-1)+5x(x-1)=(x+1)(x^4-x³+5x)
x(x+1)(x³-x²+5) ireductibil in Q

x³-x²+5 daca ar avea radacini in Q , aceste ar putea fi numai 1;-1 5 sau-5 si nu e nici una din acestea

x³-x²+5 are 1 sau 3 radacini ∈R\Q, dar nu le pot afla prin formule
sigur se mai descompune in R intr-un polinomde grad 1 si unul de grad 2, daca are o radacina, sau in 3 polinoame de  grad 1, dac are 3 radacini;
facand tabelul de variatie,(vezi anexa) cu derivata, constatam ca are doar 1 radacina irationala∈(-∞;0), de fapt ∈(-2;-1) deci in R  se mai descompune in un polinom de grad 1 si unul de grad2

d)
X^4 -X^2 +2X+2= x²(x²-1)+2(x+1)=x²(x-1)(x+1) +2(x+1)=(x+1)(x³-x²+2)
x³-x²+2=x³+1-(x²-1)=(x+1)(x²-x+1)- (x+1) (x-1)= (x+1)(x²-x+1-x+1)=(x+1)(x²-2x+2) ireductibil in R ( Δ<0)
deci

(x+1)(x³-x²+2)=(x+1)(x+1)(x²-2x+2) = (x+1)²(x²-2x+2) ireductibil in R