Question

Let p(x) be a fourth degree polynomial with coefficient of leading term 1and p(1)=p(2)=p(3)=0,then find the value ofp(0)+p(4).

Answer 1

By factor theorem, since p(1)=p(2)=p(3) = 0, (x-1)(x-2) and (x-3) are factors of the polynomial p(x).
Since the coefficient of leading term = 1, we can assume the 4th factor to be of the form (x-a).
Then p(x) = (x-a)(x-1)(x-2)(x-3)
Substitute x= 0 to get p(0) = (0-a)(0-1)(0-2)(0-3) = -a(-1)(-2)(-3) = 6a
Substitute x = 4 to get p(4) = (4-a)(4-1)(4-2)(4-3) = (4-a)(3)(2)(1) = (4-a)(6) =24-6a
So p(0) + p(4) = 6a+24-6a = 24
Answer: 24