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Find the greatest integer value of b for which the expression (9x^3 + 4x^2 + 11x + 7)/(x^2 + bx + 18) has a domain of all real numbers.

 Jul 14, 2021

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 #1
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If \(\frac{9x^3\:+\:4x^2\:+\:11x\:+\:7}{x^2\:+\:bx\:+\:18\:}\) is real, its denominator must never equal zero

The discriminant of x^2+bx+18 is b^2-72

if x^2+bx+18 doesn't equal zero, than: 

b^2-72<0

b=+/-(6 sqrt (2))

so the greatest integer is 8

 

JP

 Jul 14, 2021
 #1
avatar+208 
+1
Best Answer

If \(\frac{9x^3\:+\:4x^2\:+\:11x\:+\:7}{x^2\:+\:bx\:+\:18\:}\) is real, its denominator must never equal zero

The discriminant of x^2+bx+18 is b^2-72

if x^2+bx+18 doesn't equal zero, than: 

b^2-72<0

b=+/-(6 sqrt (2))

so the greatest integer is 8

 

JP

JKP1234567890 Jul 14, 2021

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