f ( x ) = x 2 + b x g ( x ) = 9 x 2 - 27

Question

f ( x ) = x 2 + b x g ( x ) = 9 x 2 - 27 x Functions f and g are given, and in function f , b is a constant. If f ( x ) &#183; g ( x ) = 9 x 4 - 26 x 3 - 3 x 2 , what is the value of b ?

Accepted Answer

Choice C is correct. Multiplying the given functions f and g yields f x &middot; g x = x 2 + b x 9 x 2 - 27 x . Applying the distributive property to the right-hand side of this equation yields f x &middot; g x = x 2 9 x 2 - 27 x + b x 9 x 2 - 27 x . Applying the distributive property once again to the right-hand side of this equation yields f x &middot; g x = x 2 9 x 2 - x 2 27 x + b x 9 x 2 - b x 27 x , which is equivalent to f x &middot; g x = 9 x 4 - 27 x 3 + 9 b x 3 - 27 b x 2 . Factoring out x 3 from the second and third terms yields f x &middot; g x = 9 x 4 + - 27 + 9 b x 3 - 27 b x 2 . Since the left-hand sides of f x &middot; g x = 9 x 4 + - 27 + 9 b x 3 - 27 b x 2 and f x &middot; g x = 9 x 4 - 26 x 3 - 3 x 2 are equal, it follows that - 27 + 9 b x 3 = - 26 x 3 , or - 27 + 9 b = - 26 , and - 27 b x 2 = - 3 x 2 , or - 27 b = - 3 . Adding 27 to each side of - 27 + 9 b = - 26 yields 9 b = 1 . Dividing each side of this equation by 9 yields b = 1 9 . Similarly, dividing each side of - 27 b = - 3 by - 27 yields b = 1 9 . Therefore, the value of b is 1 9 . Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Question f-x-x-