Question

$x^{10}-31x^{5}-32=0$

2)

$x^{4}-12x^{2}+27=0$

3)

$x^{8}-82x^{4}+81=0$

Answer 1

1.
$\x^{10}-31x^{5}-32=0\ x^{10}+x^5-32x^5-32=0\ x^5(x^5+1)-32(x^5+1)=0\ (x^5-32)(x^5+1)=0\ x^5-32=0 vee x^5+1=0\ x^5=32 vee x^5=-1\ x=2 vee x=-1$

2.
$\x^{4}-12x^{2}+27=0\ x^4-3x^2-9x^2+27=0\ x^2(x^2-3)-9(x^2-3)=0\ (x^2-9)(x^2-3)=0\ (x-3)(x+3)(x^2-3)=0\ x=3 vee x=-3 vee x=-sqrt{3} vee x=sqrt3$

3.
$\x^{8}-82x^{4}+81=0\ x^8-x^4-81x^4+81=0\ x^4(x^4-1)-81(x^4-1)=0\ (x^4-81)(x^4-1)=0\ (x^2-9)(x^2+9)(x^2-1)(x^2+1)=0\ (x-3)(x+3)(x^2+9)(x-1)(x+1)(x^2+1)=0\ x=3 vee x=-3 vee x=1 vee x=-1$