Paragraph: Graph of y=P(x)=a x^5+b x^4+c x^3+d x^2+e x+f , is given Question: The minimum number of real roots of the equation (P^ (x) )^2+P^ (x) P^ (x)=0 is:

Step-by-step solution

Here P^ (x)=0 at x=-2,-1,0, 1 2 P^ (x)=0 will have roots , , as (-2,-1), (-1,0) and (0, 1 2 ) [ ]+[ ]+[ ]=-2-1+0=-3 and equation P^ (x) P^ (x)=0 will have seven real roots as P^ (x)=0 has four roots and P^ (x)=0 has three roots. Its derivative (P^ (x) )^2+P^ (x) P^ (x)=0 will have 6 real roots.