SOLUTION: If -2 is a root of {{{ z^3 - 8z^2 + 9z + 58 = 0 }}}, then find the other two roots.

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Question 1181679: If -2 is a root of +z%5E3+-+8z%5E2+%2B+9z+%2B+58+=+0+, then find the other two roots.
Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
. 

In fact,  -2  IS  a root of the given polynomial, and you can check it in one line, 
substituting the value of -2 into the polynomial

    %28-2%29%5E3 - 8%2A%28-2%29%5E2 + 9%2A%28-2%29 + 58 = -8 - 32 - 18 + 58 = 0.



Therefore, the given polynomial is divisible by (z-2) without a remainer (according to the Remainder theorem)

and you can divide it using standard long division procedure 


    %28z%5E3+-+8z%5E2+%2B+9z+%2B+58%29%2F%28z-2%29 = z^2 - 10z + 29.


Find the roots of the last quadratic polynomial using the Quadratic Formula


    z%5B1%2C2%5D = %2810+%2B-+sqrt%2810%5E2+-+4%2A29%29%29%2F2 = %2810+%2B-+sqrt%28-16%29%29%2F2 = %2810+%2B-+4i%29%2F2 = 5 +- 2i.


ANSWER.  Two other roots of the given equation are  (5+2i)  and  (5-2i).

Solved.