Rozwiaz równanie. a)x5-2x3+x=0 b) x3+3x2+2x=0 c) x4=4x3+5x2 d) 6x3+9x2=3x4 e) x3+4x=-5x2 f)10x4+x3=2x2

[latex]a) \ \ x^{5}-2x^{3}+x=0 \ \ x(x^{4}-2x^{2}+1)=0 \ \ x(x^{2}-1)^{2}=0 \ \ x[(x-1)(x+1)]^{2}=0 \ \ x=0 vee x-1=0 vee x+1=0 \ \ x=0 vee x=1 vee x=-1 [/latex] [latex]b) \ \ x^{3}+3x^{2}+2x=0 \ \ x(x^{2}+3x+2)=0 \ \ x(x^{2}+2x+x+2)=0 \ \ x[x(x+2)+(x+2)]=0 \ \ x(x+2)(x+1)=0 \ \ x=0 vee x+2=0 vee x+1=0 \ \ x=0 vee x=-2 vee x=-1[/latex] [latex]c) \ \ x^{4}=4x^{3}+5x^{2} \ \ x^{4}-4x^{3}-5x^{2}=0 \ \ x^{2}(x^{2}-4x-5)=0 \ \ x^{2}(x^{2}+x-5x-5)=0 \ \ x^{2}[x(x+1)-5(x+1)]=0 \ \ x^{2}(x-5)(x+1)=0 \ \ x^{2}=0 vee x-5=0 vee x+1=0 \ \ x=0 vee x=5 vee x=-1[/latex] [latex]d) \ \ 6x^{3}+9x^{2}=3x^{4} \ \ 3x^{4}-6x^{3}-9x^{2}=0 \ \ 3x^{2}(x^{2}-2x-3)=0 \ \ 3x^{2}(x^{2}-3x+x-3)=0 \ \ 3x^{2}[x(x-3)+(x-3)]=0 \ \ 3x^{2}(x+1)(x-3)=0 \ \ 3x^{2}=0 vee x+1=0 vee x-3=0 \ \ x=0 vee x=-1 vee x=3[/latex] [latex]e) \ \ x^{3}+4x=-5x^{2} \ \ x^{3}+5x^{2}+4x=0 \ \ x(x^{2}+5x+4)=0 \ \ x(x^{2}+4x+x+4)=0 \ \ x[x(x+4)+(x+4)]=0 \ \ x(x+1)(x+4)=0 \ \ x=0 vee x+1=0 vee x+4=0 \ \ x=0 vee x=-1 vee x=-4[/latex] [latex]f) \ \ 10x^{4}+x^{3}=2x^{2} \ \ 10x^{4}+x^{3}-2x^{2}=0 \ \ x^{2}(10x^{2}+x-2)=0 \ \ x^{2}(10x^{2}+5x-4x-2)=0 \ \ x^{2}[5x(x+2)-4(x+2)]=0 \ \ x^{2}(5x-4)(x+2)=0 \ \ x^{2}=0 vee 5x-4=0 vee x+2=0 \ \ x=0 vee 5x=4 vee x=-2 \ \ x=0 vee x= frac{4}{5} vee x=-2 [/latex]