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
`(9x^3+4x^2+11x+7)/(x^2+bx+8)`. The domain of this fraction is
`x^2+bx+8\\eq 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\\eq 0` you should consider case 3, then

`b^2-32<0,\\ (b-4√(2))(b+4√(2))<0,\\ b\in (-4√(2),4√(2))`.

Therefore, the greatest integer is 5.