Question 709853
1.
{{{x^4-8x^3+19x^2-32x+60=0}}}...easiest way is to factor this completely

write {{{-8x^3}}} as {{{-3x^3-5x^3}}}, {{{19x^2}}} as {{{15x^2+4x^2}}}, and {{{-32x}}} as {{{-12x-20x}}}


{{{x^4-3x^3-5x^3-15x^2+4x^2-12x-20x-60=0}}}..group

{{{(x^4-3x^3)-(5x^3-15x^2)+(4x^2-12x)-(20x-60)=0}}}...factor out common 

{{{x^3(x-3)-5x^2(x-3)+4x(x-3)-20(x-3)=0}}}

{{{(x^3-5x^2+4x-20)(x-3)=0}}}...group

{{{((x^3-5x^2)+(4x-20))(x-3)=0}}}

{{{(x^2(x-5)+4(x-5))(x-3)=0}}}

{{{((x^2+4)(x-5))(x-3)=0}}}

{{{(x-5)(x-3)(x^2+4)=0}}}

now we can find roots

if {{{x-5=0}}}...=>...{{{x=5}}}

if {{{x-3=0}}}...=>...{{{x=3}}}

if {{{x^2+4}}}...=>...{{{x^2=-4}}}...=>...{{{x=sqrt(-4)}}}..=>...{{{x=2i}}} or {{{x=-2i}}}



2.

{{{x^5-3x^4-8x^3-8x^2-9x-5=0}}}....write {{{-3x^4}}} as {{{2x^4-5x^4}}},{{{-8x^3}}} as {{{-10x^3+2x^2}}}, {{{-8x^2}}} as {{{-10x^2+2x^2}}}, and {{{-9x}}} as {{{-10x+x}}}


{{{x^5+2x^4+x^3-5x^4-10x^3-5x^2+x^3+2x^2+x-5x^2-10x-5=0}}}...group


{{{(x^5-5x^4)+(2x^4-10x^3)+(x^3-5x^2)+(x^3-5x^2)+(2x^2-10x)+(x-5)=0}}}


{{{x^4(x-5)+2x^3(x-5)+x^2(x-5)+x^2(x-5)+2x(x-5)+(x-5)=0}}}


{{{(x^4+2x^3+x^2+x^2+2x+1)(x-5)=0}}}

{{{((x^4+x^2)+(2x^3+2x)+(x^2+1))(x-5)=0}}}

{{{(x^2(x^2+1)+2x(x^2+1)+(x^2+1))(x-5)=0}}}

{{{(x^2+2x+1)(x^2+1))(x-5)=0}}}


{{{(x+1)^2(x^2+1)(x-5)=0}}}

solutions:

if {{{(x+1)^2=0}}}...=>...{{{x+1=0}}}..=>...{{{x=-1}}}


if {{{x^2+1=0}}}...=>...{{{x^2=-1}}}..=>...{{{x=sqrt(-1)}}}...=>...{{{x=i}}} or {{{x=-i}}}

if {{{x-5=0}}}.....=>...{{{x=5}}}


{{{ graph( 600, 600, -10, 10, -300, 300, x^5-3x^4-8x^3-8x^2-9x-5) }}}