Suppose f(x) is a polynomial of degree 4 or greater such that f(1)=2, f(2)=3, and f(3)=-1. Find the remainder when f(x) is divided by (x-1)(x-2)(x-3).
+33614 "Suppose f(x) is a polynomial of degree 4 or greater such that f(1)=2, f(2)=3, and f(3)=-1. Find the remainder when f(x) is divided by (x-1)(x-2)(x-3)."
f(x) = p(x)*(x-1)(x-2)(x-3) + r(x) where r(x) is the remainder (which must be a quadratic at most). say r(x) = a*x^2 +b*x + c
2 = a(12) + b(1) + c or a + b + c = 2 (1)
3 = a(22) + 2b + c or 4a + 2b + c = 3 (2)
-1 = a(32) + 3b + c or 9a + 3b + c = -1 (3)
(2) - (1): 3a + b = 1 (4)
(3 - (2): 5a + b = -4 (5)
(5) - (4) 2a = - 5 or a = -5/2
Use this in (4) to find b = 1 - 3(-5/2) or b = 17/2
Use these two in (1) to find c. I'll leave the rest to you.
