a)
( x-4)^2(2x-7)=(x-4)(2x-7)^2
( x-4)^2(2x-7)-(x-4)(2x-7)^2=0
[(x-4)(2x-7)][(x-4)-(2x-7)]=0
(x-4)(2x-7)(x-4-2x+7)=0
(x-4)(2x-7)(3-x)=0
-(x-4)(2x-7)(x-3)=0|*(-1)
(x-4)(2x-7)(x-3)=0
x-4=0\vee 2x=7\vee x-3=0
x=4\vee x=\frac{7}{2}\vee x=3
x=4\vee x=3,5 \vee x=3
b)
2x^5 - 3x^3 - 2x^2 +3=0
2x^3(x^2-\frac{3}{2})-2(x^2-\frac{3}{2})=0
(x^2-\frac{3}{2})(2x^3-2)=0
(x-\sqrt{\frac{3}{2}})(x+\sqrt{\frac{3}{2}})*2(x^3-1)=0
2(x-\frac{\sqrt3}{2})(x+\frac{\sqrt3}{2})(x^3-1)=0
x-\frac{\sqrt3}{2}=0\vee x+\frac{\sqrt3}{2}=0 \vee x^3=1
x=\frac{\sqrt3}{2} \vee x=-\frac{\sqrt3}{2}\vee x=1
c)
x^3(2x-5)-x^3=0
2x^4-5x^3-x^3=0
2x^4-6x^3=0
2x^3(x-3)=0
2x^3=0\vee x-3=0
x=0\vee x=3
d)
x^5 - 2x^3+3x^2-6=0
x^3(x^2-2)-3(x^2-2)=0
(x^2-2)(x^3-3)=0
[x^2-(\sqrt2)^2] (x^3-3)=0
(x-\sqrt3)(x+\sqrt3)(x^3-3)=0
x-\sqrt3=0\vee x+\sqrt3=0 \vee x^3-3=0
x=\sqrt3\vee x=-\sqrt3\vee x=\sqrt[3]3