pre calculus simplify the following polynomials (2x^3 - x^2 y +6x -3y)/(2x -y)

To simplify the polynomial \((2x^3 - x^2y + 6x - 3y)/(2x - y)\), we can use polynomial long division.

1. **Set up the division**:
   Divide \(2x^3 - x^2y + 6x - 3y\) by \(2x - y\).

2. **Divide the leading terms**:
   The leading term of the numerator is \(2x^3\) and the leading term of the denominator is \(2x\). Dividing these gives \(x^2\).

3. **Multiply and subtract**:
   Multiply \(x^2\) by \(2x - y\):
   \[
   x^2(2x - y) = 2x^3 - x^2y
   \]
   Subtract this from the original polynomial:
   \[
   (2x^3 - x^2y + 6x - 3y) - (2x^3 - x^2y) = 6x - 3y
   \]

4. **Repeat with the new polynomial**:
   Now divide \(6x - 3y\) by \(2x - y\). The leading term \(6x\) divided by \(2x\) gives \(3\).

5. **Multiply and subtract again**:
   Multiply \(3\) by \(2x - y\):
   \[
   3(2x - y) = 6x - 3y
   \]
   Subtract:
   \[
   (6x - 3y) - (6x - 3y) = 0
   \]

Since the remainder is \(0\), the final result of the division is:
\[
\frac{2x^3 - x^2y + 6x - 3y}{2x - y} = x^2 + 3
\]

Thus, the simplified form is:
\[
x^2 + 3
\]

### Questions
1. Do you want to see the steps of polynomial long division in detail?
2. Would you like examples of similar problems?
3. Are you interested in how this might apply to functions or graphs?
4. Do you need clarification on any specific step in the simplification?
5. Would you like to explore further topics in polynomial functions?

### Tip
Always check if the polynomial can be factored before performing long division, as it can simplify the process.

(2x^3 - x^2 y + 6x - 3y)/(2x - y) simplify the following polynomials

Discover how to simplify the polynomial (2x^3 - x^2y + 6x - 3y)/(2x - y) through polynomial long division. This detailed guide provides step-by-step instructions and explanations suitable for students in Grades 9-11.

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