f(x) = x^5 - 2x^4 - 2x^3 + x^2 + 4x + 4 has (x-2) as a factor. Determine which of the following is true about the set of roots for f(x).  a) 2 is a single root with 4 more real and 0 complex....

f(x) = x^5 - 2x^4 - 2x^3 + x^2 + 4x + 4 has (x-2) as a factor.

Determine which of the following is true about the set of roots for f(x). 

a) 2 is a single root with 4 more real and 0 complex.

b) 2 is a single root with 2 more real and 2 complex.

c) 2 is a single root with no real and 4 complex.

d) 2 is a double root with 3 more real and 0 complex.

e) 2 is a double root with 1 more real and 2 complex.

f) 2 is a triple root with no root and 2 complex.

g) 2 is a triple root with 2 more real and 0 complex. 

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`f(x) = x^5 - 2x^4 - 2x^3 + x^2 + 4x + 4`

`=x^5-2x^4-2x^3+4x^2-3x^2+6x-2x+4`

`=(x-2)(x^4-2x^2-3x-2)`

`=(x-2)(x^4-2x^3+2x^3-4x^2+2x^2-4x+x-2)`

`=(x-2)(x-2)(x^3+2x^2+2x+1)`

`=(x-2)^2(x^3+x^2+x^2+x+x+1)`

`=(x-2)^2(x+1)(x^2+x+1)`

`x^2+x+1=x^2+x+1/4+3/4`

`=(x+1/2)^2-((sqrt(3)i)^2)/(2^2)`

`=(x+1/2+(isqrt(3))/2)(x+1/2-(isqrt(3))/2)`

Thus

`f(x)=(x-2)^2(x+1)(x+1/2+(isqrt(3))/2)(x+1/2-(isqrt(3))/2)`

2 is double zero

-1 is real single zero

two complex zero.

answer is (e)

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