Question

Descompuneti polinomul x^4+3x^2+2 in factori ireductibili

Answer 1

Explicație pas cu pas:

x^4+3x^2+2=0

x^2=y

y²+3y+2=0

Δ=9-8=1

y1=(-3+1)/2=-2/2=-1

y2=(-3-1)/2=-4/2=-2

x²=-1⇒x∈∅

x²=-2 ⇒ x∈∅

Nu poate fi descompus.

x^4-3x²+2=0

x²=y

y²-3y+2=0

Δ=1

x1=(3+1)/2=4/2=2

x2=(3-1)/2=2/2=1

x²=2

x²=1

deci:

x^4-3x²+2=(x²-2)(x²-1)=(x-√2)(x+√2)(x-1)(x+1)

Cred ca poate asa ai avut in vedere.

Bafta!

Answer 2

$x^4+3x^2+2 = 0\\ \\ \text{Observam ca }i\text{ este solutie.} \\ \\ \Rightarrow x_1 = i,\quad x_2 = -i. \\ \\ x_1+x_2+x_3+x_4 =0 \Rightarrow i-i+x_3+x_4 = 0 \Rightarrow \\ \Rightarrow x_3+x_4 = 0\\\\ x_1x_2x_3x_4 = 2 \Rightarrow i\cdot(-i)\cdot x_3x_4=2 \Rightarrow x_3x_4 = 2\\ \\ t^2-(x_3+x_4)t+x_3x_4 = 0 \\ t^2+2 = 0 \\ \Rightarrow t^2 = -2 \Rightarrow x_{3,4} = \pm i\sqrt{2}$

$\Rightarrow x^4+3x^2+2 = (x-i)(x+i)(x-i\sqrt 2)(x+i\sqrt 2)$