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]}

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