+0  
 
+1
40
2
avatar+742 

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
avatar+92814 
+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

 

 

cool cool cool

CPhill  Nov 29, 2018
 #2
avatar+742 
+1

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

ant101  Nov 29, 2018

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