The ratio of the coefficient of x^3 to the term independent of x in the expansion of (2 x+ 1 x^2 )¹² is

Step-by-step solution

The general term in the expansion of (2x + 1 x^2 )¹² is given by T_ r+1 = ¹²C_ r (2x)^ 12-r ( 1 x^2 )^ r = ¹²C_ r 2^ 12-r x^ 12-r x^ -2r = ¹²C_ r 2^ 12-r x^ 12-3r . To find the coefficient of x^3 , set the exponent 12 - 3r = 3 , which gives 3r = 9 , so r = 3 . The coefficient of x^3 is ¹²C₃ 2¹²⁻³ = ¹²C₃ 2⁹ = 12 11 10 3 2 1 2^9 = 220 512 . To find the term independent of x , set the exponent 12 - 3r = 0 , which gives 3r = 12 , so r = 4 . The term independent of x is ¹²C₄ 2¹²⁻⁴ = ¹²C₄ 2⁸ = 12 11 10 9 4 3 2 1 2^8 = 49