I have a problem with simplifying the polynomial. In the first time, I see that this polynomial is quite simple, but when I'm trying, I realized that this polynomial isn't as easy as I saw.
Here is the problem:
\begin{eqnarray} \left(x^3-x^2y-2xy^2-x^2+8xy-6x\right) \leq0\\\end{eqnarray}
My attempt: First I isolated x, so that x is outside this equation:
\begin{eqnarray} \left(x(x^2-xy-2y^2-x+8y-6)\right) \leq 0\ \end{eqnarray}
Then, I'm trying to find the variables that can I modify, such that the equation can be simplify like this: \begin{eqnarray} \left(x(x^2-2xy+xy-2y^2-x+8y-6)\right) \leq 0\ \end{eqnarray}
Until this equation, I cannot go any further.
Can anybody help me to continue this problem and find its solution?
And, also what is the easiest way to simplify this kind of this problem?
Thanks
Look for $a,b,c,d$ such that: $$x^2−xy−2y^2−x+8y−6 = (x+a y+b)(x+cy+d) \\ = x^2+(a+c)xy+(ac)y^2+(b+d)x+(ad+bc)y+bd$$
So solve: $\left[\begin{array}{rcl}a+c&=&-1\\ac&=&-2\\b+d&=&-1\\ad+bd&=&+8\\bd&=&-6\end{array}\right] \implies a= 1,b=-3, c=-2, d=2$
$$\therefore x^2−xy−2y^2−x+8y−6 = (x+ y-3)(x-2y+2)$$