Find P0 where Px is the degree 5 polynomial that is zero at

Find P(0) where P(x) is the degree 5 polynomial that is zero at x = 1, 2, 3, 4 and satisfies P(5) = -24.

Solution

P(x)=(x-1)(x-2)(x-3)(x-4)(Ax+B)

P(5)=-24=(5-1)(5-2)(5-3)(5-4)(5A+B)=-24=24*(5A+B)

So, 5A+B=-1

B=-5A-1

P(x)=(x-1)(x-2)(x-3)(x-4)(Ax-5A-1)=

P(0)=-24(5A+1)