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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).

 Oct 12, 2020
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"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.

 Oct 12, 2020

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