How do you divide #(n^3 + 2n^2 - n - 2) # by #(n^2 - 1)#?

We have that

#(n^3 + 2n^2 - n - 2)/(n^2 - 1)=(n^3-n+2(n^2-1))/(n^2-1)= (n(n^2-1)+2(n^2-1))/(n^2-1)=((n^2-1)*(n+2))/(n^2-1)=n+2#

To divide the polynomial (n^3 + 2n^2 - n - 2) by (n^2 - 1), we perform polynomial long division or synthetic division.

Using polynomial long division:

1. Divide the first term of the dividend by the first term of the divisor.
2. Multiply the entire divisor by the result obtained in step 1.
3. Subtract the result obtained in step 2 from the dividend.
4. Repeat steps 1-3 until the degree of the remainder is less than the degree of the divisor.

Using synthetic division:

1. Write down the coefficients of the dividend and divisor.
2. Change the sign of the constant term of the divisor.
3. Perform synthetic division as usual.
4. Write down the quotient obtained.

After performing polynomial long division or synthetic division, we obtain the quotient (n + 3) and the remainder (2n + 1). Therefore, the result of dividing (n^3 + 2n^2 - n - 2) by (n^2 - 1) is (n + 3) with a remainder of (2n + 1).