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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.

 May 3, 2022
 #1
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If the function has a domain of all real numbers, then dividing by x^2 + bx + 18 always "makes sense".

When does that "make sense"? It is when x^2 + bx + 18 ≠ 0 for any real number x.

Then either x^2 + bx + 18 > 0 for any real number x or x^2 + bx + 18 < 0 for any real number x.

But the latter is not possible, because the graph of y = x^2 + bx + 18 opens upwards.

 

Then x^2 + bx + 18 > 0 for any real number x.

(i.e., there are no x-intercepts)

Since there are no x-intercepts, \(\Delta = b^2 - 4(1)(18) < 0\).

You can solve the inequality and find the greatest integer b satisfying this inequality.

 May 3, 2022

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