1: Suppose f is a polynomial such that f(0) = 47, f(1) = 32, f(2) = -13, and f(3)=16. What is the sum of the coefficients of f?
Let us suppose that we have this form
P(x) = ax^3 + bx^2 + cx + d where d = 47
So we have that
a + b + c + 47 = 32 ⇒ a + b + c = -15
8a + 4b + 2c + 47 = -13 ⇒ 8a + 4b + 2c = -60
27a + 9b + 3c + 47 = 16 ⇒ 27a + 9b + 3c = -31
Multiply the first equation by -2 and add it to the second equation
Multiply the first equation by -3 and add it to the third equation
6a + 2b = -30 ⇒ 3a + b = - 15
24a + 6b = 14 ⇒ 12a + 3b = 7
Multiply the first equation by -3 and add it to the second
3a = 52 ⇒ a = 52/3
3(52/3) + b = -15 ⇒ 52 + b = -15 ⇒ b = -67
(52/3) + (-67) + c = -15 ⇒ c = -15 - 52/3 + 67 ⇒ c = 104/3
So.... the sum of the coefficients is
52/3 - 67 + 104/3 = -15