The number of polynomials of the form x 3 + a x 2 + b x + c ∀ a , b , c ∈ 1 ,2 , 3 , . . . . . , 10 which are divisible by x 2 + 1 is equal to

Step-by-step solution

f x = x 3 + a x 2 + b x + c is divisible by x 2 + 1 or x + i x - i So, f i = 0 and f - i = 0 f i = 0 ⇒ i 3 + a i 2 + b i + c = 0 ⇒ - i - a + b i + c = 0 ⇒ c - a + i b - 1 = 0 ⇒ c = a , b = 1 f - i = 0 ⇒ - i 3 + a - i 2 + b - i + c = 0 ⇒ i - a - i b + c = 0 ⇒ c - a - i b - 1 = 0 ⇒ c = a ,   b = 1 b = 1 and c = a ⇒ 10 polynomials are possible