Question

Descompuneti in factori ireductibili :
a) x^3-11x^2+30x
b) x^3-14x^2+48x
c)x^2(x^2-10)+9
d) x^2(x^2-13)+36
e) (x^2-x)(x^2-x+7)-18
f) (x^2+2x)(x^2+2X-4)-5
g) 4(x+5)^2-(x-1)^2
h)(x2+3x+1)2-(x+3)^2
i) (x^2+3x+2)^2-(x-1)2

Answer 1

a) x^3-11x^2+30x=x(x^2-11x+30)
x^2-11x+30 ec grad 2 cu solutiile 5 si 6
Δ=121-120=1, x1=(11+1)/2, x2=(11-1)/2
deci x^2-11x+30=(x-5)(x-6)
deci
x^3-11x^2+30x=x(x-5)(x-6)
b)
x^3-14x^2+48x=x(x^2-14x+48)
(x^2-14x+48) ec grad 2 cu solutiile 8 si 6
Δ=196-192=4, x1=(14+2)/2=8, x2=12/2=6 
(x^2-14x+48) =(x-8)(x-6)
x^3-14x^2+48x=x(x^2-14x+48)=x(x+8)(x+6)

Answer 2

a) x^3-11x^2+30x=x(x^2-11x+30)=x(x-5)(x-6)

b) x^3-14x^2+48x=x(x^2-14x+48)=x(x-6)(x-8)

c) x^2(x^2-10)+9=x^4 -10x^2 +9=(x^2-1)(x^2-9)=(x-1)(x+1)(x-9)(x+9)

d) x^2(x^2-13)+36=x^4 -12x^2 +36=(x^2 -6)^2

e) (x^2-x)(x^2-x+7)-18=(x^2-x=y)=y(y+7)-18=y^2 +7y-18=
(y+9)(y-2)=(x^2-x+9)(x^2-x-2)=(x^2-x+9)(x+1)(x-2)

f)  (x^2+2x)(x^2+2X-4)-5=(y=x^2+2x)=y(y-4)-5=y^2-4y-5=(y+1)(y-5)=
(x^2+2x+1)(x^2+2x-5)=(x+1)^2 *(x^2+2x-5)

g) 4(x+5)^2-(x-1)^2=(4x+20-x+1)(4x+20+x-1)=(3x+21)(5x-19)=3(x+7)(3x+19)

h) (x^2+3x+1)^2-(x+3)^2=(x^2+3x+1-x-3)(x^2+3x+1+x+3)=(x^2+2x-2)(x^2+4x+4)=(x^2+2x-2)(x+2)^2

i) (x^2+3x+2)^2-(x-1)^2=(x^2+3x+2-x+1)(x^2+3x+2+x-1)=(x^2+2x+3)(x^2+4x+1)