Question

(Wielomiany) rozwiąż równanie:
x³-5x-4=0

x³-3x+2=0

x⁴-7x²+6x=0

Answer 1

A)
x³-x-4x-4 = x(x²-1)-4(x+1) = x(x+1)(x-1)-4(x+1)=(x+1)(x²-x-4)
x+1=0  v  x²-x-4=0
x=-1  v Δ=1+4*4=17
x=(1-√17)/2 v x=(1+√17)/2
Odp. x∈{-1, (1-√17)/2, (1+√17)/2}

b)  x³ -3x+2=x³-x-2x+2=x(x²-1)-2(x-1)=x(x+1)(x-1)-2(x-1)=
(x-1)(x²+x-2)=(x-1)(x²-x+2x-2)=(x-1)*[x(x-1)+2(x-1)]=
(x-1)² * (x+2)=0
x-1=0  v  x+2=0
x=1  v  x=-2

c)  x⁴-7x²+6x= x(x³-x-6x+6)=x*[x(x²-1)-6(x-1)]=x*[x(x+1)(x-1)-6(x-1)]=
x(x-1)(x²+x-6)=x(x-1)(x²+3x-2x-6)=x(x-1)*[x(x+3)-2(x+3)]=
x(x-1)(x-2)(x+3)=0
x=0  v  x-1=0  v  x-2=0  v  x+3=0
Odp. x∈{-3,0,1,2}