Question

The polynomial p(x)=ax2+bx-c is divisible by the polynomial g(x)=x2+bx+c.c×0 if?​

Answer 1

f(x)=ax3+bx−c

g(x)=x2+bx+c

f(x) divides g(x)means roots of g(x) are also roots of f(x)

Let p,q be the roots of g(x) and p,q,r be the roots of f(x)

g(x):

Sum of the roots = Coefficient of x =p+q=−b

Product of the roots =pq=c

f(x):

Sum of the roots =Coefficient of x2=p+q+r=0

⟹−b+r=0⟹r=b

Product of the roots = pqr=ca

Substituting r=b and pq=c in the above:

⟹c.b=ca⟹⟹ab=1

Ans: ab=1