Why Does the Root Sum Calculation for a Polynomial's Derivative Confuse Many?

[fayeshin]

The question is :"The polynomial 3x⁵-250x³+735x is interesting because it has the maximum possible number of relative extrema and points of inflection at integer lattice points for a quintic polynomial. What is the sum of the x-coordinates of these points?"

I think the answer is 0, but it's wrong.

My solution is :"The first derivative is 15x⁴-750x²+735, whose roots are -7,-1,1 and 7. The second derivative is 60x³-1500x,whose roots are 0,-5,5. Then, the sum is 0."

The HMMT solution is "The first derivative is 15x⁴-750x²+735, whose roots sum to 750/15=50. The second derivative is 60x³-1500x,whose roots sum to 1500/60=25, for a grand total of 75."

I really can't understand how it get the sum 50 and 25...Please help me.

Thanks...

 

[HallsofIvy]

The only thing I can say is that they appear to be talking about the sum of x² where x is a zero of the polynomial:
$$15x^4-750x^2+735= 15(x^2- 1)(x^2- 49)$$
so x²= 1 and 49 which add to 50 and
$$60x3-1500x= 60 x(x^2- 25)$$
so x²= 0 and 25 which add to 25.

 

[fayeshin]

The only thing I can say is that they appear to be talking about the sum of x² where x is a zero of the polynomial:
$$15x^4-750x^2+735= 15(x^2- 1)(x^2- 49)$$
so x²= 1 and 49 which add to 50 and
$$60x3-1500x= 60 x(x^2- 25)$$
so x²= 0 and 25 which add to 25.

i think so...
but since we have +7 and -7 and so on...why isn't it 150?

 

[fayeshin]

thank you all the way...