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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 \(b?\)



 Nov 29, 2018

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




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



cool cool cool

 Nov 29, 2018

Oh, I get it! Just have to look at the denominator! Thank you, CPhill! 

ant101  Nov 29, 2018

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