SOLUTION: Find all zeros, both real and complex. x^3 + 3x^2 + 4x - 8. Thanks

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Question 320120: Find all zeros, both real and complex. x^3 + 3x^2 + 4x - 8.
Thanks
Found 2 solutions by Fombitz, solver91311:
Answer by Fombitz(13828) About Me  (Show Source):
You can put this solution on YOUR website!
+graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C++x%5E3+%2B+3x%5E2+%2B+4x-8%29
Looks like x=1 is a zero.
Verify using the equation.
1%2B3%2B4-8=0
Yes, so x-1 is a factor.
Use polynomial long division to find the remaining quadratic factor.
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First term: x%5E2
x%5E2%28x-1%29=x%5E3-x%5E2
Subtract from the original polynomial to get the remainder,
%28x%5E3%2B3x%5E2%2B4x-8%29-%28x%5E3-x%5E2%29=4x%5E2%2B4x-8
Next term: 4x
4x%28x-1%29=4x%5E2-4x
Subtract from the first remainder to get the next remainder,
%284x%5E2%2B4x-8%29-%284x%5E2-4x%29=8x-8
Final term:8
8%28x-1%29=8x-8
Subtract from the second remainder to get the next remainder,
%288x-8%29-%288x-8%29=0
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%28x%5E3%2B3x%5E2%2B4x-8%29%2F%28x-1%29=x%5E2%2B4x%2B8
Solve for the remaining zeros by completing the square.
x%5E2%2B4x%2B8=0
x%5E2%2B4x%2B4%2B8-4=0
%28x%2B2%29%5E2%2B4=0
%28x%2B2%29%5E2=-4
x%2B2=0+%2B-+2i
highlight%28x=-2+%2B-+2i%29
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x=1
x=-2%2B2i
x=-2-2i

Answer by solver91311(16897) About Me  (Show Source):
You can put this solution on YOUR website!




Use the rational root theorem. Possible rational roots for this polynomial function are

Use synthetic division to check the eight possible roots. If you need a review of synthetic division see www.purplemath.com

In fact, there is one rational root for this function. Once you find it by synthetic division, you will have a quadratic factor. Use the quadratic equation to determine the other two zeros of the original function.


John