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Advanced Math / Equivalent expressions Difficulty: Hard

Which of the following expressions has a factor of $x plus 2 b$ , where $b$ is a positive integer constant?

$3 x^{2} + 7 x + 14 b$

$3 x^{2} + 28 x + 14 b$

$3 x^{2} + 42 x + 14 b$

$3 x^{2} + 49 x + 14 b$

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Explanation

Choice D is correct. Since each choice has a term of $3 x squared$ , which can be written as $left parenthesis 3 x right parenthesis left parenthesis x right parenthesis$ , and each choice has a term of $14 b$ , which can be written as $left parenthesis 7 right parenthesis left parenthesis 2 b right parenthesis$ , the expression that has a factor of $x plus 2 b$ , where $b$ is a positive integer constant, can be represented as $(3 x + 7) (x + 2 b)$ . Using the distributive property of multiplication, this expression is equivalent to $3 x (x + 2 b) + 7 (x + 2 b)$ , or $3 x^{2} + 6 x b + 7 x + 14 b$ . Combining the x-terms in this expression yields $3 x^{2} + (7 + 6 b) x + 14 b$ . It follows that the coefficient of the x-term is equal to $7 plus 6 b$ . Thus, from the given choices, $7 plus 6 b$ must be equal to $7$ , $28$ , $42$ , or $49$ . Therefore, $6 b$ must be equal to $0$ , $21$ , $35$ , or $42$ , respectively, and $b$ must be equal to $StartFraction 0 Over 6 EndFraction$ , $StartFraction 21 Over 6 EndFraction$ , $StartFraction 35 Over 6 EndFraction$ , or $StartFraction 42 Over 6 EndFraction$ , respectively. Of these four values of $b$ , only $StartFraction 42 Over 6 EndFraction$ , or $7$ , is a positive integer. It follows that $7 plus 6 b$ must be equal to $49$ because this is the only choice for which the value of $b$ is a positive integer constant. Therefore, the expression that has a factor of $x plus 2 b$ is $3 x^{2} + 49 x + 14 b$ .

Choice A is incorrect. If this expression has a factor of $x plus 2 b$ , then the value of $b$ is $0$ , which isn't positive.

Choice B is incorrect. If this expression has a factor of $x plus 2 b$ , then the value of $b$ is $StartFraction 21 Over 6 EndFraction$ , which isn't an integer.

Choice C is incorrect. If this expression has a factor of $x plus 2 b$ , then the value of $b$ is $StartFraction 35 Over 6 EndFraction$ , which isn't an integer.