SOLUTION: When {{{ax^3-x^2+3x+b}}} is divided by x - 2, the remainder is 59. When it is divided by x + 1, the remainder is

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Question 1204344: When is divided by x - 2, the remainder is 59. When it is divided by x + 1, the remainder is - 1. Find the values of a and b.
Answer by math_tutor2020(3817) (Show Source):
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Remainder Theorem:
If P(x) is divided over (x-k), then P(k) is the remainder.

f(x) = ax^3 - x^2 + 3x + b
f(2) = a(2)^3 - (2)^2 + 3(2) + b
f(2) = 8a + b + 2
f(2) = 59 because of the remainder theorem
8a + b + 2 = 59
b = 59-2-8a
b = 57-8a

f(x) = ax^3 - x^2 + 3x + b
f(-1) = a(-1)^3 - (-1)^2 + 3(-1) + b
f(-1) = -a + b - 4
f(-1) = -1 because of the remainder theorem
-a + b - 4 = -1
-a + (57-8a) - 4 = -1
-9a + 53 = -1
-9a = -1-53
-9a = -54
a = -54/(-9)
a = 6

Then,
b = 57-8a
b = 57 - 8*6
b = 57 - 48
b = 9

SOLUTION: When {{{ax^3-x^2+3x+b}}} is divided by x - 2, the remainder is 59. When it is divided by x + 1, the remainder is - 1. Find the values of a and b.