SOLUTION: Solve the equation {{{x^3+2x^2+x+2=0}}}, if -2 is a root.

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Question 175796This question is from textbook
: Solve the equation x%5E3%2B2x%5E2%2Bx%2B2=0, if -2 is a root. This question is from textbook

Answer by EMStelley(208) About Me  (Show Source):
You can put this solution on YOUR website!
If you know that -2 is a root, that means that %28x-%28-2%29%29=x%2B2 is a factor of the polynomial. In other words, when you divide x%5E3%2B2x%5E2%2Bx%2B2 by x%2B2, the remainder will by 0. The easiest way to do this is with synthetic division. Or you could try factoring. I'm not sure of the best way to type synthetic division in here, so I will leave that up to you. Once you complete the division, you get that
x%5E3%2B2x%5E2%2Bx%2B2=%28x%2B2%29%28x%5E2%2B1%29
So to solve, we get that
x%2B2=0 and x%5E2%2B1=0
The first results in x=-2 which we already knew about. So we need to solve the second. All that needs to be done is to subtract the 1 from both side and take the square root.
x=%2Bsqrt%28-1%29 and x=-sqrt%28-1%29
Since the square root of -1 is i, we have that x=i and x=-i