How do you list all possible roots and find all factors and zeroes of #x^4-x^3+14x^2-16x-32#?

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Answer 1 · 2018-07-05T13:45:30

Please see the explanation below

The function is

#P(x)=x^4-x^3+14x^2-16x-32#

#=x^4-x^3-color(red)(2x^2)+14x^2+color(red)(2x^2)-16x-32#

#=(x^4-x^3-2x^2)+(16x^2-16x-32)#

#=x^2(x^2-x-2)+16(x^2-x-2)#

#=(x^2+16)(x^2-x-2)#

#=(x^2+16)(x+1)(x-2)#

The real roots are #(x=-1)# and #(x=2)# and the imaginary roots are #(x=4i)# and #(x=-4i)#

graph{x^4-x^3+14x^2-16x-32 [-36.53, 36.53, -18.27, 18.28]}