The expression 4 x 2 + b x - 45 , where b is a constant, can be rewritten as

Question

The expression 4 x 2 + b x - 45 , where b is a constant, can be rewritten as h x + k x + j , where h , k , and j are integer constants. Which of the following must be an integer?

Accepted Answer

Choice D is correct. It's given that 4 x 2 + b x - 45 can be rewritten as h x + k x + j . The expression h x + k x + j can be rewritten as h x 2 + j h x + k x + k j , or h x 2 + j h + k x + k j . Therefore, h x 2 + j h + k x + k j is equivalent to 4 x 2 + b x - 45 . It follows that k j = - 45 . Dividing each side of this equation by k yields j = - 45 k . Since j is an integer, - 45 k must be an integer. Therefore, 45 k must also be an integer. 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 C is incorrect and may result from conceptual or calculation errors.

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