Question 334835
Graphing the function, {{{x=4}}} is the only real root.
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{{{graph(300,300,-2,6,-4,4,x^3-8x^2+36x-80)}}}
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Use polynomial long division to find the remaining quadratic equation,
{{{(x^3-8x^2+36x-80)/(x-4)}}}
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First term : {{{x^2}}}
{{{x^2(x-4)=x^3-4x^2}}}
Subtract this product from the original polynomial to get the remainder,
{{{(x^3-8x^2+36x-80)-(x^3-4x^2)=-4x^2+36x-80}}}
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Second term : {{{-4x}}}
{{{-4x(x-4)= -4x^2+16x}}}
Subtract this product from the remainder to get the new remainder,
{{{( -4x^2+36x-80)-(-4x^2+16x)=20x-80}}}.
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Final term : {{{20}}}
{{{20(x-4)=20x-80}}}
Subtract this product from the remainder to get the new remainder,
{{{( 20x-80)-(20x-80)=0}}}.
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Gather the terms,
{{{(x^3-8x^2+36x-80)/(x-4)=(x^2-4x+20)}}}
Complete the square to get the remaining roots,
{{{x^2-4x+20=0}}}
{{{x^2-4x+4+16=0}}}
{{{(x-2)^2=-16}}}
{{{x-2=0 +- 4i}}}
{{{x=2 +- 4i}}}