How do you find all the zeros of f(x)=x5+4x4+5x3?

1 Answer
George C.
Feb 26, 2016

f(x)=x5+4x4+5x3=x3(x+2−i)(x+2+i)

Hence the zeros of f(x) are x=0 and x=−2±i.

Explanation:

First separate out the common factor x3.

Then complete the square and use the difference of squares identity:

a2−b2=(a−b)(a+b)

with a=x+2 and b=i, as follows:

f(x)=x5+4x4+5x3

=x3(x2+4x+5)

=x3(x2+4x+4+1)

=x3((x+2)2−i2)

=x3((x+2)−i)((x+2)+i)

=x3(x+2−i)(x+2+i)

Hence the zeros of f(x) are x=0 and x=−2±i.

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