Consider fraction [tex] \dfrac{9x^3+4x^2+11x+7}{x^2+bx+8} [/tex]. The domain of this fraction is [tex] x^2+bx+8\neq 0 [/tex].
Find the discriminant of the the equation [tex] x^2+bx+8=0 [/tex]:
[tex] D=b^2-4\cdot 1\cdot 8=b^2-32 [/tex].
There are choices:
1. If D>0, there are two solutions of the equation;
2. D=0, there is only unique solution of the equation;
3. D<0, there are no solutions.
If [tex] x^2+bx+8\neq 0 [/tex] you should consider case 3, then
[tex] b^2-32<0,\\ (b-4\sqrt{2})(b+4\sqrt{2})<0,\\ b\in (-4\sqrt{2},4\sqrt{2}) [/tex].
Therefore, the greatest integer is 5.
