+884 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!
+128475 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
