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
=> 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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We'll note the polynomial P(X) = aX^3+bX^2+cX+d.
We'll write the reminder theorem, when P(x) is divided by (X-1):
P(1)=3
We'll write the reminder theorem, when P(x) is divided by (X+1):
P(-1)=3
We'll write the reminder theorem, when P(x) is divided by (X+2):
P(-2)=3
From these facts, we notice that the reminder of the division of P(x) to the product of polynomials (X-1)(X+1)(X+2) is also 3.
We'll write the reminder theorem:
aX^3+bX^2+cX+d=(X-1)(X+1)(X+2) + 3
We'll remove the brackets:
aX^3+bX^2+cX+d=(X^2-1)(X+2) +3
aX^3+bX^2+cX+d = X^3 + 2X^2-X -2 + 3
We'll combine like terms and we'll get:
aX^3 + bX^2 + cX + d = X^3 + 2X^2 - X + 1
P(X) = X^3 + 2X^2 - X + 1
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