Question

Proszę o pomoc na teraz!!!!

Answer 1

a)

[tex]w(x)=2x^3-7x^2+7x-2=2x^3-2-7x^2+7x=2(x^3-1)-7x(x-1)=\\=2(x-1)(x^2+x+1)-7x(x-1)=(x-1)(2x^2+2x+2-7x)=\\=(x-1)(2x^2-5x+2)=(x-1)(2x^2-4x-x+2)=(x-1)[2x(x-2)-(x-2)]=\\=(x-1)(x-2)(2x-1)[/tex]

b)

[tex]w(x)=2x^3-3x^2-3x+2=2x^3+2-3x^2-3x=2(x^3+1)-3x(x+1)=\\=2(x+1)(x^2-x+1)-3x(x+1)=(x+1)(2x^2-2x+2-3x)=\\=(x+1)(2x^2-5x+2)=(x+1)(2x^2-4x-x+2)=(x+1)[2x(x-2)-(x-2)]=\\=(x+1)(x-2)(2x-1)[/tex]

c)

[tex]w(x)=27x^3-6x^2-2x+1=27x^3+1-6x^2-2x=(27x^3+1)-2x(3x+1)=\\=(3x+1)(9x^2-3x+1)-2x(3x+1)=(3x+1)(9x^2-3x+1-2x)=\\=(3x+1)(9x^2-5x+1)[/tex]

Drugi nawias nie jest rozkładalny, bo

[tex]\Delta=(-5)^2-4*9*1=25-36=-11 < 0[/tex]

d)

[tex]w(x)=3x^4-10x^3+10x-3=3x^4-3-10x^3+10x=3(x^4-1)-10x(x^2-1)=\\=3(x^2-1)(x^2+1)-10x(x^2-1)=(x^2-1)(3x^2+3-10x)=\\=(x-1)(x+1)(3x^2-9x-x+3)=(x-1)(x+1)[3x(x-3)-(x-3)]=\\=(x-1)(x+1)(x-3)(3x-1)[/tex]

e)

[tex]w(x)=x^4+3x^3-6x-4=x^4-4+3x^3-6x=(x^4-4)+3x(x^2-2)=\\=(x^2-2)(x^2+2)+3x(x^2-2)=(x^2-2)(x^2+2+3x)=\\=(x-\sqrt2)(x+\sqrt2)(x^2+3x+2)=(x-\sqrt2)(x+\sqrt2)(x^2+2x+x+2)=\\=(x-\sqrt2)(x+\sqrt2)[x(x+2)+(x+2)]=(x-\sqrt2)(x+\sqrt2)(x+2)(x+1)[/tex]

f)

[tex]w(x)=4x^4+2x^3+x-1=4x^4-1+2x^3+x=(4x^4-1)+x(2x^2+1)=\\=(2x^2-1)(2x^2+1)+x(2x^2+1)=(2x^2+1)(2x^2-1+x)=\\=(2x^2+1)(2x^2+x-1)=(2x^2+1)(x^2+x+x^2-1)=\\=(2x^2+1)[x(x+1)+(x^2-1)]=\\=(2x^2+1)[x(x+1)+(x-1)(x+1)]=(2x^2+1)(x+1)(x+x-1)=\\=(2x^2+1)(x+1)(2x-1)[/tex]