One of the factors of 2 x 3 + 42 x 2 + 208 x is x + b ,

Question

One of the factors of 2 x 3 + 42 x 2 + 208 x is x + b , where b is a positive constant. What is the smallest possible value of b ?

Accepted Answer

The correct answer is 8 . Since each term of the given expression, 2 x 3 + 42 x 2 + 208 x , has a factor of 2 x , the expression can be rewritten as 2 x x 2 + 2 x 21 x + 2 x 104 , or 2 x x 2 + 21 x + 104 . Since the values 8 and 13 have a sum of 21 and a product of 104 , the expression x 2 + 21 x + 104 can be factored as x + 8 x + 13 . Therefore, the given expression can be factored as 2 x x + 8 x + 13 . It follows that the factors of the given expression are 2 , x , x + 8 , and x + 13 . Of these factors, only x + 8 and x + 13 are of the form x + b , where b is a positive constant. Therefore, the possible values of b are 8 and 13 . Thus, the smallest possible value of b is 8 .

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