Which of the following expressions has a factor of x + 2 b , where b is a positive integer

Question

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

Accepted Answer

Choice D is correct. Since each choice has a term of 3 x 2 , which can be written as 3 x x , and each choice has a term of 14 b , which can be written as 7 2 b , the expression that has a factor of x + 2 b , where b is a positive integer constant, can be represented as 3 x + 7 x + 2 b . Using the distributive property of multipliion, 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 + 6 b . Thus, from the given choices, 7 + 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 0 6 , 21 6 , 35 6 , or 42 6 , respectively. Of these four values of b , only 42 6 , or 7 , is a positive integer. It follows that 7 + 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 + 2 b is 3 x 2 + 49 x + 14 b . Choice A is incorrect. If this expression has a factor of x + 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 + 2 b , then the value of b is 21 6 , which isn't an integer. Choice C is incorrect. If this expression has a factor of x + 2 b , then the value of b is 35 6 , which isn't an integer.

Question 629129