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We are tasked with performing polynomial long division on the expression:

\[
\frac{2x^3 - x^2 + 9x}{x^2 - x + 2}
\]

Let's walk through the steps of polynomial long division.

1. **First Step:** Divide the leading term of the dividend by the leading term of the divisor:
   \[
   \frac{2x^3}{x^2} = 2x
   \]
   This gives the first term of the quotient, \(2x\).

2. **Second Step:** Multiply the divisor by this first term \(2x\) and subtract from the original polynomial:
   \[
   (2x^3 - x^2 + 9x) - (2x \cdot (x^2 - x + 2)) = (2x^3 - x^2 + 9x) - (2x^3 - 2x^2 + 4x)
   \]
   Simplifying:
   \[
   2x^3 - x^2 + 9x - (2x^3 - 2x^2 + 4x) = (2x^3 - 2x^3) + (-x^2 + 2x^2) + (9x - 4x) = x^2 + 5x
   \]

3. **Third Step:** Repeat the process by dividing the leading term of the new polynomial \(x^2 + 5x\) by the leading term of the divisor \(x^2 - x + 2\):
   \[
   \frac{x^2}{x^2} = 1
   \]
   This gives the next term in the quotient, which is \(1\).

4. **Fourth Step:** Multiply the divisor by \(1\) and subtract from the polynomial:
   \[
   (x^2 + 5x) - (1 \cdot (x^2 - x + 2)) = (x^2 + 5x) - (x^2 - x + 2)
   \]
   Simplifying:
   \[
   (x^2 - x^2) + (5x + x) - 2 = 6x - 2
   \]

5. **Final Step:** Since \(6x - 2\) has a degree lower than the divisor, it is the remainder.

Thus, the quotient is \(2x + 1\) and the remainder is \(6x - 2\). The final result can be written as:

\[
\frac{2x^3 - x^2 + 9x}{x^2 - x + 2} = 2x + 1 + \frac{6x - 2}{x^2 - x + 2}
\]

Now the polynomial \(p(x)\) is expressed as:

\[
p(x) = d(x)q(x) + r(x)
\]

where:
- \(d(x) = x^2 - x + 2\) (divisor),
- \(q(x) = 2x + 1\) (quotient),
- \(r(x) = 6x - 2\) (remainder).

Would you like further details on any of the steps? Here are some related questions:

1. How does polynomial long division work when the divisor has a higher degree?
2. Can you explain how synthetic division differs from polynomial long division?
3. What happens if the remainder is 0 in polynomial division?
4. How can polynomial division be used to find asymptotes of rational functions?
5. How do we handle division of polynomials when the divisor is not monic (leading coefficient ≠ 1)?

**Tip:** When performing polynomial long division, always align similar terms vertically to simplify calculations.

Use polynomial long division to perform the indicated division: (2x^3 - x^2 + 9x) ÷ (x^2 - x + 2). Then, write the polynomial p(x) = 2x^3 - x^2 + 9x in the form d(x)q(x) + r(x).

Learn how to perform polynomial long division to divide 2x^3 - x^2 + 9x by x^2 - x + 2. Follow a detailed step-by-step guide to express the polynomial in the form p(x) = d(x)q(x) + r(x). Ideal for high school algebra students in Grades 9-12.

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