let f(x) be a polynomial function. if f(x) is divided by x-1,x+1 and x+2 then remainders are 5,3and 2 respectively. whwn f(x) is divided by x^3+2x^2-x-2 then remainder is:

To find the remainder when f(x) is divided by x^3 + 2x^2 - x - 2, we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by x - a, the remainder is equal to f(a).

Given that f(x) divided by x - 1 gives a remainder of 5, f(1) = 5.
Similarly, f(x) divided by x + 1 gives a remainder of 3, f(-1) = 3.
And f(x) divided by x + 2 gives a remainder of 2, f(-2) = 2.

Now, we can use these values to find the remainder when f(x) is divided by x^3 + 2x^2 - x - 2.

Substituting x = 1 into f(x), we get f(1) = 5.
Substituting x = -1 into f(x), we get f(-1) = 3.
Substituting x = -2 into f(x), we get f(-2) = 2.

To find the remainder when f(x) is divided by x^3 + 2x^2 - x - 2, we can construct a system of equations using these values.

Let R be the remainder when f(x) is divided by x^3 + 2x^2 - x - 2. The remainder theorem gives us the following equations:

R = f(1)
R = f(-1)
R = f(-2)

Solving this system of equations, we find that the remainder R is equal to 3.

Therefore, the remainder when f(x) is divided by x^3 + 2x^2 - x - 2 is 3.