<s>[INST] What is wrong with my polynomial division? $x^3-2x^2-5x+6$ $$x^3-2x^2-5x+6$$
I want to get the solutions of this. I did a polynomial division.
First, I know that $(x+1)$ is a factor since $1^3-2\cdot1^2-5\cdot1+6 = 0$
So my division goes like this:
$$..........x^2-3x-2$$
$$(x+1)|\overline{x^3-2x^2-5x+6}$$
$$x^3+x^2$$
$$......\overline{0-3x^2}-5x$$
$$........-3x^2-3x$$
$$...............\overline{0-2x}+6$$
$$....................2x-2$$
$$....................\overline{0 + 8}$$
(Sorry for the improvised formatting. Ignore any dots you see there.)
So I get, at the end, the quadratic $x^2-3x-2+8=x^2-3x+6$
However, $\triangle = (-3)^2-4(1)(6)=9-4(6)=9-24=-15$
Therefore, there are no solutions since $\triangle$ is negative.
But I definitely did something wrong, since I do know that the solutions are $1,-2,3$
What did I do wrong?
 [/INST] $(x+1)$ isn't a factor; you discovered that $1$ is a zero of your polynomial, hence $(x-1)$ is a factor.
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