When f(x)=2x^3+ax^2+bx+c is divided by (x^2-1) the remainder is x+1. If (x-2) is a factor of f(x), find the values of the constants a,b and c.

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To find the values of the constants a, b, and c, we need to use the Remainder Theorem. According to the Remainder Theorem, when a polynomial f(x) is divided by (x - k), the remainder is equal to f(k).

Given that the remainder when f(x) is divided by (x^2 - 1) is (x + 1), we can substitute (x^2 - 1) into f(x). This will give us the remainder function:

f(x) = (x^2 - 1) * q(x) + (x + 1)

where q(x) is the quotient of the division.

Now let's substitute (x = 2) into the equation since (x - 2) is a factor of f(x). Based on the Remainder Theorem, the remainder at (x = 2) will be zero if (x - 2) is a factor.

Substituting (x = 2) into the equation gives us:
0 = (2^2 - 1) * q(2) + (2 + 1)
0 = 3 * q(2) + 3
0 = 3(q(2) + 1)

From the equation above, we can see that (q(2) + 1) must be zero in order for (x - 2) to be a factor. Therefore, q(2) must be -1. Hence, q(x) can be written as:

q(x) = (x - 2) * r(x) - 1

where r(x) is another quotient.

Now, substitute (x^2 - 1) into f(x) along with the expression for q(x):

f(x) = (x^2 - 1) * [(x - 2) * r(x) - 1] + (x + 1)
f(x) = (x^2 - 1)(x - 2) * r(x) - (x^2 - 1) + (x + 1)
f(x) = (x^3 - 2x^2 - x + 2x^2 - 4x - 2) * r(x) - x^2 + 1 + x + 1
f(x) = (x^3 - 5x - 2) * r(x) - x^2 + x - 1

Now, compare the above expression of f(x) with the given expression of f(x) = 2x^3 + ax^2 + bx + c. In order for both expressions to be equivalent, the coefficients of corresponding terms must be equal. Therefore, we can identify the values of the constants a, b, and c as follows:

a = -1, b = 1, c = -1

Therefore, the values of a, b, and c are -1, 1, and -1 respectively.