Question

Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers.

Answer 1

Consider fraction $\dfrac{9x^3+4x^2+11x+7}{x^2+bx+8}$. The domain of this fraction is $x^2+bx+8\neq 0$.

Find the discriminant of the the equation $x^2+bx+8=0$:

$D=b^2-4\cdot 1\cdot 8=b^2-32$.

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 $x^2+bx+8\neq 0$ you should consider case 3, then
$b^2-32$.

Therefore, the greatest integer is 5.