Odpowiedź:
a)
[tex]W(x)=(x+3)(x^2+6)[/tex]
b)
[tex]W(x)=(x-2)(3x^2+4)[/tex]
c)
[tex]W(x)=(x-1)(3x-1)(3x+1)[/tex]
d)
[tex]W(x)=5(x+3)(x-\frac{\sqrt{5}}{5})(x+\frac{\sqrt{5}}{5})[/tex]
e)
[tex]W(x)=x(x+2)(x-3)(x^2+3x+9)[/tex]
Szczegółowe wyjaśnienie:
a)
[tex]W(x)=x^3+3x^2+6x+18=\\\\=x^2(x+3)+6x+18=\\\\=x^2(x+3)+6(x+3)=\\\\=(x+3)(x^2+6)[/tex]
b)
[tex]W(x)=3x^3-6x^2+4x-8=\\\\=3x^2(x-2)+4x-8=\\\\=3x^2(x-2)+4(x-2)=\\\\=(x-2)(3x^2+4)[/tex]
c)
[tex]W(x)=9x^3-9x^2-x+1=\\\\=9x^2(x-1)-x+1=\\\\=9x^2(x-1)-(x-1)=\\\\=(x-1)(9x^2-1)=\\\\=(x-1)(3x-1)(3x+1)[/tex]
d)
[tex]W(x)=5x^3+15x^2-x-3=\\\\=5x^2(x+3)-x-3=\\\\=5x^2(x+3)-(x+3)=\\\\=(x+3)(5x^2-1)=\\\\=5(x+3)(x^2-\frac{1}{5})=\\\\=5(x+3)(x-\frac{\sqrt{5}}{5})(x+\frac{\sqrt{5}}{5})[/tex]
e)
[tex]W(x)=x^5+2x^4-27x^2-54x=\\\\=x^4(x+2)-27x^2-54x=\\\\=x^4(x+2)-27x(x+2)=\\\\=(x+2)(x^4-27x)=\\\\=x(x+2)(x^3-27)=\\\\=x(x+2)(x-3)(x^2+3x+9)[/tex]
[tex]\Delta=3^2-4\cdot 1\cdot 9=9-36=-27<0[/tex]
