The polynomial ax^2+bx+c is divisible by x-1 and has a remainder of 2 when divided by x+1 and has a remainder of 8 when divided by x-2. Find the value of a, b and c
+129840 Using synthetic division
1 [ a b c ]
a b + a
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a b + a a + b + c = 2
2 [ a b c ]
2a 2b + 4a
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a b +2a 4a + 2b + c = 8
We have this system
a + b + c = 2
4a + 2b + c = 8
We will have infinite solutions....however....if we assume a monic poynomial (a = 1) then we have a solution
1 + b + c = 2 ⇒ b + c = 1 ⇒ c = 1 - b (1)
4 + 2b + c = 8 (2)
Sub (1) into (2)
4 + 2b + 1 - b = 8
5 + b = 8
b = 3
c = 1 - 3 = -2
a =1 b = 3 c = -2
