It looks like you uploaded a list of rational functions. Let’s simplify each of the given expressions.

1. $\frac{4x^3 - 3x + 2}{x + 2}$

To simplify this, we use polynomial division.

2. $\frac{6x^3 - 17x^2 + 11x - 2}{x - 2}$

Again, use polynomial division to divide the numerator by $x - 2$.

3. $\frac{12x^3 - 11x^2 + 9x + 18}{4x + 3}$

This one requires division of the cubic polynomial by $4x + 3$.

4. $\frac{2x^3 + 5x^2 - 7x + 2}{2x - 1}$

Perform polynomial division of the cubic polynomial by $2x - 1$.

5. $\frac{4x^4 + 17x^3 - 8x^2 - 15x + 3}{4x + 1}$

Divide the quartic polynomial by $4x + 1$.

Let me work through the polynomial divisions for these. I’ll start with the first one and proceed to the others.Here are the results after simplifying each of the given rational functions using polynomial division:

1. $\frac{4x^3 - 3x + 2}{x + 2} = 4x^2 - 8x + 13 + \frac{-24}{x + 2}$

2. $\frac{6x^3 - 17x^2 + 11x - 2}{x - 2} = 6x^2 - 5x + 1$

3. $\frac{12x^3 - 11x^2 + 9x + 18}{4x + 3} = 3x^2 - 5x + 6$

4. $\frac{2x^3 + 5x^2 - 7x + 2}{2x - 1} = x^2 + 3x - 2$

5. $\frac{4x^4 + 17x^3 - 8x^2 - 15x + 3}{4x + 1} = x^3 + 4x^2 - 3x - 3 + \frac{6}{4x + 1}$

Let me know if you need further clarification or more details on the steps.

Related Questions:

1. How is synthetic division different from long polynomial division?
2. What happens when a polynomial cannot be exactly divided by another polynomial?
3. Can polynomial division result in a remainder that is not a polynomial?
4. How do you interpret the remainder in polynomial division?
5. What are other methods to simplify rational functions?

Tip:

Always check for possible factorization of polynomials before proceeding with division. It can sometimes simplify the problem.