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Equivalent expressions SAT question

Math · Advanced Math · Hard · Answer it here, explanation included.

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?

ExplanationShow

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 $j = \frac{- 45}{k}$ . Since $j$ is an integer, $- \frac{45}{k}$ must be an integer. Therefore, $\frac{45}{k}$ 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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