# Okay so i need to find the zeros of the following function `x7+5x6-20x5-139x4-161x3+326x2+900x+600` ``I found all the real zeros they are, -2, 2, 5, -5 if that helps you at all what i need is the...

Okay so i need to find the zeros of the following function `x7+5x6-20x5-139x4-161x3+326x2+900x+600`

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I found all the real zeros they are, -2, 2, 5, -5 if that helps you at all what i need is the imaginary zeros can you show me how you did it at as well thanks

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### 1 Answer

`x^7 + 5x^6 - 20x^5 - 139x^4 -161x^3 + 326x^2 + 900x +600= 0`

`` Given the real roots are : -2, 2, 5, and -5.

Then we know that the factors are:

(x+2) (x-2)(x-5)(x+5).

Now we will find the product of the factors.

`(x-2)(x+2)(x-5)(x+5)= (x^2 -4)(x^2-25) = (x^4 -29x^2 +100` )

Then, (`x^4 -29x^2 + 100` ) is a factor for the polynomial.

Then we will divide the polynomial by the factor (`x^4-29x^2+100` ) and we get (`x^3 +5x^2 + 9x +6` )

Now we will need to factor (`x^3 +5x^2 +9x +6` ).

==> `(x^3 + 5x^2 +9x+6)= (x+2)(x^2 +3x +3)`

`` ==> Now we will find the roots for (`x^2 + 3x +3` ).

==> `x1= (-3+sqrt(9-12))/2 = (-3+sqrt(3)i)/2` .

==> `x2= (-3-sqrt3*i)/2` .

Then, the imaginary roots are:

`x= { (-3+sqrt3*i)/2 , (-3-sqrt3*i)/2 }`