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[nati22]

a) W(x)=x^3+8
b) W(x)=(x+3)^2 -(2x+1)^2
c) W(x)=(2x-4)^3-(2-x)^3
d) W(x)=x^3-1
e)W(x)=-5x^5+30x^4-45x^3
f)W(x)=3x^3-15x^2-6x+30

[hhtp]

\(\displaystyle{ a) W(x)=x^3+8=x^3+2^3=(x+2)(x^2-2x+4)
\newline
b) W(x)=(x+3)^2 -(2x+1)^2=
[(x+3)-(2x+1)]\cdot [(x+3)+(2x+1)]=
(x+3-2x-1)(x+3+2x+1)=(2-x)(3x+4)
\newline
c) W(x)=(2x-4)^3-(2-x)^3=
[(2x-4)-(2-x)]\cdot[(2x-4)^2+(2x-4)(2-x)+(2-x)^2]=
(2x-4-2+x)(4x^2-16x+16+4x-8-2x^2+4x+4-4x+x^2)=
(3x-6)(3x^2-12x+12)=
(3x-6)3(x^2-4x+4)=
3(3x-6)(x-2)^2\newline
d) W(x)=x^3-1=
x^3-1^3=(x-1)(x^2+x+1)\newline

e)W(x)=-5x^5+30x^4-45x^3
-5x^3(x^2-6x+9)=
-5x^3(x-3)^2\newline

f)W(x)=3x^3-15x^2-6x+30
3x^2(x-5)-6(x-5)=
(x-5)(3x^2-6)=
(x-5)3(x^2-2)=
3(x-5)(x-\sqrt2)(x+\sqrt2)}\)