Total number of polynomials of the form x^3+a x^2+b x+c that are divisible by x^2+1 , where a, b, c 1,2,3, , 10 is equal to

Step-by-step solution

(x^2+1 )=(x-2)(x+2) Let f(x)=x^3+a x^2+b x+c ; a, b, c 1,2, , 10 f(x) is divisible by x^2+1 iff f(i)=0, f(-i)=0 i^3+a i^2+b i+c=0 array rrr & -i-a+b i+c=0 & a=c, b=1 array and -i^3+a i^2-b i+c=0 array rr & i-a-b i+c=0 & a=c, b=1 array Total number of selection of triplets (a, b, c)=(1,1,1),(2,1,2), ,(10,1,10) Hence, number of such polynomials =10 .