how to simplify 2x^3-x-3/(x+1)

To simplify the expression (2x^3 - x - 3) / (x + 1), you can use polynomial long division or synthetic division. I will explain how to use polynomial long division to simplify the expression.

1. Begin by dividing the term with the highest power in the numerator (2x^3) by the term with the highest power in the denominator (x). The result is 2x^2.

_________
x + 1 | 2x^3 - x - 3

2x^2

2. Now, multiply the divisor (x + 1) by the quotient (2x^2) and write the product below the dividend.

_________
x + 1 | 2x^3 - x - 3
2x^3 + 2x^2

3. Next, subtract the product from the dividend and bring down the next term.

_________
x + 1 | 2x^3 - x - 3
2x^3 + 2x^2
___________________
-3x^2 - x

4. Repeat the process until all terms have been divided. Now divide the new expression (-3x^2 - x) by the divisor (x + 1), which results in -3x.

_________
x + 1 | 2x^3 - x - 3
2x^3 + 2x^2
___________________
-3x^2 - x
-3x^2 - 3x
_________________________
2x - 3

5. Now bring down the constant term (-3) and divide it by the divisor (x + 1), resulting in -3.

_________
x + 1 | 2x^3 - x - 3
2x^3 + 2x^2
___________________
-3x^2 - x
-3x^2 - 3x
_________________________
2x - 3
2x - 2
_________________________
-1

6. At this point, the remainder is -1. The simplified expression is the quotient (2x^2 - 3x + 2) plus the remainder (-1) divided by the divisor (x + 1), which gives:

2x^2 - 3x + 2 - 1 / (x + 1)

Therefore, the simplified expression is: 2x^2 - 3x + 1 / (x + 1).