Question 143275
{{{P(x)=x^3-5x^2+16x-80}}} Start with the given function



{{{0=x^3-5x^2+16x-80}}} Plug in {{{P(x)=0}}}



{{{0=(x^3-5x^2)+(16x-80)}}} Group like terms



{{{0=x^2(x-5)+16(x-5)}}} Factor out the GCF {{{x^2}}} out of the first group. Factor out the GCF {{{16}}} out of the second group



{{{0=(x^2+16)(x-5)}}} Since we have the common term {{{x-5}}}, we can combine like terms




Now set each factor equal to zero:


{{{x^2+16=0}}} or  {{{x-5=0}}} 


Now solve for x for each factor:


{{{x^2=-16}}} or  {{{x=5}}} 



{{{x=0+-sqrt(-16)}}} or  {{{x=5}}}



Take the square root of -16 to get 4i and -4i


{{{x=4i}}}, {{{x=-4i}}}, or {{{x=5}}}




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Answer:

So our answers are



{{{x=4i}}}, {{{x=-4i}}}, or {{{x=5}}}