SOLUTION: Find all zeros of the polynomial functions. Show the process. x^4 - 2x^3 + 6x^2 - 18x -27

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find all zeros of the polynomial functions. Show the process. x^4 - 2x^3 + 6x^2 - 18x -27      Log On


   

Question 983908: Find all zeros of the polynomial functions. Show the process.
x^4 - 2x^3 + 6x^2 - 18x -27
Found 2 solutions by stanbon, josgarithmetic:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Find all zeros of the polynomial functions. Show the process.
x^4 - 2x^3 + 6x^2 - 18x -27
----
Graph to see the zeroes::
graph%28400%2C400%2C-10%2C10%2C-10%2C10%2Cx%5E4-2x%5E3%2B6x%5E2-18x-27%29
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Use the zero at x = 3
3)....1....-2....6....-18....-27
......1.....1....9....9....|..0
Use the zero at x = -1
------
-1)....1....1....9....9
.......1....0....9...|..0
Factor:: x^2 + 9 = 0
------
Imaginary zeroes:: x = 3i or x = -3i
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Ans: Zeroes at x = 3 OR x = -1 OR x = 3i OR x = -3i
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Cheers,
Stan H.
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Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
Possible roots to try are PLUS AND MINUS OF 1,3,9,27.

Use synthetic division to test. 

-1    |    1    -2    6    -18    -27
      |         -1    3    -9      27
      |___________________________________
           1    -3    9    -27     0


-1    |    1    -3    9    -27
      |         -1    4    -13
      |___________________________________
           1    -4    13    -40



-3    |    1    -3    9    -27
      |         -3   18    -81
      |___________________________________
           1    -6   27    -108


1     |     1    -3    9    -27
      |           1    -2    7
      |___________________________
            1    -2    7     -20


3      |     1    -3    9    -27
       |           3    0     27
       |__________________________
             1     0    9     0


So far, two roots and the quadratic binomial:
highlight_green%28%28x%2B1%29%28x-3%29%28x%5E2%2B9%29%29.

The last two roots are imaginary, x=-i%2Asqrt%283%29 and x=i%2Asqrt%283%29.