Question

Descompuneti in factori ireductibili polinomul:
a)x³+2x²(x-2)-8=
b)x^6-4x³+4=
c)x^5-x³+5x²+5x=
d)x⁴-x²+2x+2=​

Answer 1

Răspuns:

Explicație pas cu pas:

a) x^3 + 2x^2(x-2) - 8 = (x^3 - 8) + 2x^2(x-2) = (x-2)(x^2 + 2x + 4) + 2x^2(x-2) =

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

= (x - 2) [x(x+2) + (x+2)] = (x-2)(x+2)(x+1)

b) x^6 -4x^3+4 = (x^3 - 2)^2

c) x^5 - x^3 + 5x^2 + 5x = x^3(x^2 - 1) + 5x(x+1) = x^3(x-1)(x+1) + 5x(x+1) =

= (x+1)[x^2(x-1) + 5x] = x(x+1)[x(x-1) + 5 ] = x(x+1)(x^2 - x + 5)

d) x^4 - x^2 + 2x + 2 = x^2 ( x^2 - 1) + 2(x+1) = x^2(x-1)(x+1) + 2(x+1) =

= (x+1)[x^2(x-1) + 2] = (x+1)[x^3 - x^2 +2 ]

Am folosit identitatile:

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

a^2 - b^2 = (a-b)(a+b)