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.

I know the interval notation, but how would you solve for \(b?\)

Thanks!

ant101
Nov 29, 2018

#1**+2 **

We only need to worry about the denominator being 0

We need to find the largest "b" that would give no real roots to the polynomial in the denominator

To do this....set the discriminant = 0

b^2 - 4(1)(8) = 0

b^2 - 32 = 32

b^2 = 32

Take the square root and b ≈ 5.66

This means that if b = 5, we will have no real zeros in the denominator because the discriminant will be < 0

So....the greatest integer value of b = 5

To get a feel for this look at the graphs of x^2 + 6x + 8 and x^2 + 5x + 8

https://www.desmos.com/calculator/iselamtz4y

Note that the first will have real zeros, but the second will not since it never intersects the x axis

CPhill
Nov 29, 2018