Let P(x) and Q(x) are two different polynomials with real coefficients satisfying the conditions: (i) a and b are the roots of P(x) and Q(x) respectively. (ii) P(b) Q(a)>0 . Then:

Step-by-step solution

Given, P(a)=0Q(b)=0 Consider, aligned & R(x)=P(x)-Q(x) & R(a)=-Q(a) & R(b)=P(b) & R(a) R(b)=-Q(a) P(b) < 0 aligned using I.V.T. R(c)=0 for some c P(c)-Q(c)=0 for some c | ly consider R(x)=P(x)-2 Q(x)R(a)=-2 Q(a) ; R(b)=P(b) R(a) R(b)=-2 Q(a) P(b) < 0 R(c)=0 for some c (using I.V.T.) P(c)-2 Q(c)=0 for some c (c)