SOLUTION: Find all the roots of the following polynomial in the complex number system. f(x)=x^3-8x^2+36x-80

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: Find all the roots of the following polynomial in the complex number system. f(x)=x^3-8x^2+36x-80       Log On


   

Question 334835: Find all the roots of the following polynomial in the complex number system.
f(x)=x^3-8x^2+36x-80
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Graphing the function, x=4 is the only real root.
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graph%28300%2C300%2C-2%2C6%2C-4%2C4%2Cx%5E3-8x%5E2%2B36x-80%29
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Use polynomial long division to find the remaining quadratic equation,
%28x%5E3-8x%5E2%2B36x-80%29%2F%28x-4%29
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First term : x%5E2
x%5E2%28x-4%29=x%5E3-4x%5E2
Subtract this product from the original polynomial to get the remainder,
%28x%5E3-8x%5E2%2B36x-80%29-%28x%5E3-4x%5E2%29=-4x%5E2%2B36x-80
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Second term : -4x
-4x%28x-4%29=+-4x%5E2%2B16x
Subtract this product from the remainder to get the new remainder,
%28+-4x%5E2%2B36x-80%29-%28-4x%5E2%2B16x%29=20x-80.
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Final term : 20
20%28x-4%29=20x-80
Subtract this product from the remainder to get the new remainder,
%28+20x-80%29-%2820x-80%29=0.
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Gather the terms,
%28x%5E3-8x%5E2%2B36x-80%29%2F%28x-4%29=%28x%5E2-4x%2B20%29
Complete the square to get the remaining roots,
x%5E2-4x%2B20=0
x%5E2-4x%2B4%2B16=0
%28x-2%29%5E2=-16
x-2=0+%2B-+4i
x=2+%2B-+4i