You
can do synthetic division by complex roots. It's a little tricky, but it can be done.
First, divide $\displaystyle x^4 - 9x^3 + 21x^2 +21x -130$ by
3 - 2i. We know that the remainder will be zero:
Code:
3 - 2i| 1 -9 21 21 -130
------- 3-2i -22+6i 9+20i 130
--------------------------------
1 -6-2i -1+6i 30+20i 0
Now, let's divide the quotient by
3 + 2i, the conjugate of 3 - 2i:
Code:
3 + 2i| 1 -6-2i -1+6i 30+20i
------- 3+2i -9-6i -30-20i
---------------------------
1 -3 -10 0
So you're left with the quadratic
$\displaystyle x^2 - 3x - 10 = 0$
which factors into
$\displaystyle (x + 2)(x - 5) = 0$
The two remaining roots are -2 and 5.
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