sat suite question viewer

Advanced Math / Equivalent expressions Difficulty: Hard

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?

$\frac{b}{h}$

$\frac{b}{k}$

$StartFraction 45 Over h EndFraction$

$StartFraction 45 Over k EndFraction$

Back question 33 of 102 Next

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102

Explanation

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, $StartFraction 45 Over k EndFraction$ 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.