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# Nice Question

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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 $$b?$$

Thanks!

Nov 29, 2018

#1
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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

Nov 29, 2018
#2
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Oh, I get it! Just have to look at the denominator! Thank you, CPhill!

ant101  Nov 29, 2018