Let p (x) be a 4th degree polynomial with leading coefficient as 1 and 1,2,3 as its roots.Find p (0)+p (6)

Question

I took the equation as x^4+ax^3+bx^2+cx+d . But I couldn't solve as 1 got only 3 equations i.e. p(1)=0,p (2)=0,p (3)=0 and for solving for 4 variables a,b,c,d atleast 4 equations are required.

Answer 1

We can write p(x)=(x-a)(x-1)(x-2)(x-3) to include the roots and the coefficient of x^4 equalling 1. The fourth root is a.

p(0)=-a(-1)(-2)(-3)=6a (the constant term); p(6)=(6-a)(5)(4)(3)=360-60a.

p(0)+p(6)=360-60a+6a=360-54a.

We are told all the roots, assumedly, so there must be a duplicated root and a=1, 2 or 3. So p(0)=6, 12 or 18.

Therefore, p(0)+p(6)=36, -288 or -612.