What is polynomial ax^3+bx^2+cx+d if divided by (x-1),(x+1),(x+2), the reminder is always 3?

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The polynomial ax^3 + bx^2 + cx + d is the product of the terms (x-1), (x+1) and (x+2) to which the remainder 3 is added.

(x - 1)(x +1)(x + 2) +3

=> (x^2 - 1)(x + 2) +3

=> x^3 - x + 2x^2 - 2 +3

=>...

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The polynomial ax^3 + bx^2 + cx + d is the product of the terms (x-1), (x+1) and (x+2) to which the remainder 3 is added.

(x - 1)(x +1)(x + 2) +3

=> (x^2 - 1)(x + 2) +3

=> x^3 - x + 2x^2 - 2 +3

=> x^3 + 2x^2 - x + 1

Now x^3 + 2x^2 - x + 1 = ax^3 + bx^2 + cx + d

Equating the coefficients of x, x^2 , x^3 and the numeric term we get a = 1 , b = 2 , c = -1 and d = 1

The polynomial is x^3 + 2x^2 - x + 1

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