Question

if the polynomial f(X)= x cube + b x minus C is divisible by the polynomial g(X) =X square + bx + c then find the value of a b

Answer 1

Answer:

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