\left(x^{2}-4x+11\right)\div\left(x-3\right)

To divide \(\left(x^{2}-4x+11\right)\) by \(\left(x-3\right)\), we can use polynomial long division.

### Step-by-Step Solution:

1. **Divide the first term of the dividend by the first term of the divisor:**

   \[
   \frac{x^2}{x} = x
   \]

2. **Multiply the entire divisor by the result from step 1:**

   \[
   x \cdot (x - 3) = x^2 - 3x
   \]

3. **Subtract the result from the original polynomial:**

   \[
   (x^2 - 4x + 11) - (x^2 - 3x) = -x + 11
   \]

4. **Repeat the process with the new polynomial:**

   - Divide the first term of the new polynomial by the first term of the divisor:
     \[
     \frac{-x}{x} = -1
     \]
   - Multiply the entire divisor by this result:
     \[
     -1 \cdot (x - 3) = -x + 3
     \]
   - Subtract this from the new polynomial:
     \[
     (-x + 11) - (-x + 3) = 8
     \]

Since no further division is possible, the quotient is:

\[
\boxed{x - 1}
\]

And the remainder is:

\[
\boxed{8}
\]

So the final answer can be written as:

\[
\frac{x^2 - 4x + 11}{x - 3} = x - 1 + \frac{8}{x - 3}
\]

Would you like more details on any of these steps, or do you have any other questions?

---

Here are five related questions you might find useful:

1. How do you interpret the remainder in a polynomial division?
2. What happens when you divide a polynomial by a linear factor and the remainder is zero?
3. Can you use synthetic division to divide \(\left(x^{2}-4x+11\right)\) by \(\left(x-3\right)\)?
4. How does the division of polynomials differ when the divisor is not a linear polynomial?
5. What is the relationship between the divisor, quotient, and remainder in polynomial division?

**Tip:** After performing polynomial division, you can always check your result by multiplying the quotient by the divisor and adding the remainder to see if you get the original polynomial.

Learn how to divide the polynomial x^2 - 4x + 11 by the linear polynomial x - 3 using polynomial long division. Includes step-by-step solution, quotient, and remainder analysis. Suitable for high school students studying polynomial division.

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