The required ploynomial has a highest degree of 3.
When it is divided by (x + 2) and (x^2 - 1) the remainder is 3.
So we get the polynomial as (x + 2)(x^2 - 1) + 3
=> x^3 + 2x^2 - x - 2 + 3
=> x^3 + 2x^2 - x + 1
The required polynomial is x^3 + 2x^2 - x + 1
The polynomial is not determined, yet. It will be determined, when all it's coefficients will be known.
We'll write the polynomial whose leading term is ax^3 as:
P = ax^3 + bx^2 + cx + d
Since we know the reminder that we've get when P is divided by given binomials, we'll apply the reminder theorem.
We notice that x^2 - 1 is a difference of 2 squares and it represents the product:
x^2 - 1 = (x-1)(x+1)
We'll write the reminder theorem, when P(x) is divided by (X-1):
We'll write the reminder theorem, when P(x) is divided by (X+1):
We'll write the reminder theorem, when P(x) is divided by (X+2):
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^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
The requested polynomial is: P(X) = X^3 + 2X^2 - X + 1.