OG详解-OG2 数学1 Q20

The correct answer is 8. Since each term of the given expression, 2x³ + 42x² +208x, has a factor of 2x, the expression can be rewritten as 2x(x² )+2x(21x)+2x(104), or 2x(x² +21x +104). Since the values 8  and 13  have a sum of 21 and a product of  104, the expression x² +21x +104  can be factored as (x +8)(x +13). Therefore, the given expression can be factored as 2x(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.