Let f x and g x be two real polynomials of degree 2 and 1 respectively. If f g x = 8 x 2 - 2 x , and g f x = 4 x 2 + 6 x + 1 , then the value of f 2 + g 2 is ______.

Step-by-step solution

Given, f g x = 8 x 2 - 2 x and g f x = 4 x 2 + 6 x + 1 Let f x = c x 2 + d x + e and g x = a x + b So, f g x = c a x + b 2 + d a x + b + e = 8 x 2 - 2 x and g f x = a c x 2 + d x + e + b = 4 x 2 + 6 x + 1 On comparing both side we get, a = 2 ,   b = - 1 ,   c = 2 ,   d = 3 ,   e = 1 So, g x = 2 x - 1 ⇒ g 2 = 3 &   f x = 2 x 2 + 3 x + 1 f 2 = 8 + 6 + 1 = 15 . So, f 2 + g 2 = 18