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Equivalent expressions SAT question

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

Which of the following expressions is(are) a factor of $3 x^{2} + 20 x - 63$ ?

$x - 9$
$3 x - 7$

ExplanationShow

Choice B is correct. The given expression can be factored by first finding two values whose sum is $20$ and whose product is $3 (- 63)$ , or $- 189$ . Those two values are $27$ and $- 7$ . It follows that the given expression can be rewritten as $3 x^{2} + 27 x - 7 x - 63$ . Since the first two terms of this expression have a common factor of $3 x$ and the last two terms of this expression have a common factor of $- 7$ , this expression can be rewritten as $3 x (x + 9) - 7 (x + 9)$ . Since the two terms of this expression have a common factor of $(x + 9)$ , it can be rewritten as $(3 x - 7) (x + 9)$ . Therefore, expression II, $3 x - 7$ , is a factor of $3 x^{2} + 20 x - 63$ , but expression I, $x - 9$ , is not a factor of $3 x^{2} + 20 x - 63$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

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