\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'll use polynomial long division.

### Step 1: Divide the first term of the numerator by the first term of the denominator.
\[
\frac{x^{2}}{x} = x
\]

### Step 2: Multiply \(x\) by the entire divisor \(\left(x - 3\right)\).
\[
x \cdot (x - 3) = x^{2} - 3x
\]

### Step 3: Subtract the result from the original polynomial.
\[
(x^{2} - 4x + 11) - (x^{2} - 3x) = -x + 11
\]

### Step 4: Repeat the process with the new polynomial \(-x + 11\).
\[
\frac{-x}{x} = -1
\]

### Step 5: Multiply \(-1\) by the divisor \(\left(x - 3\right)\).
\[
-1 \cdot (x - 3) = -x + 3
\]

### Step 6: Subtract the result from \(-x + 11\).
\[
(-x + 11) - (-x + 3) = 8
\]

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

So, the result of the division is \(x - 1 + \frac{8}{x - 3}\).

Would you like to see more details, or do you have any questions?

Here are 5 related questions for practice:
1. Divide \(x^2 + 5x + 6\) by \(x + 2\).
2. Divide \(2x^2 - 3x + 1\) by \(x - 1\).
3. Simplify \(\frac{x^3 - 2x^2 + x - 2}{x - 1}\).
4. What is the quotient when \(3x^2 - 12x + 20\) is divided by \(x - 2\)?
5. Divide \(4x^2 + 8x + 5\) by \(2x + 3\).

**Tip:** Always align like terms vertically when performing polynomial long division to avoid mistakes.

Learn how to divide the polynomial (x^2 - 4x + 11) by (x - 3) using polynomial long division. Follow a detailed step-by-step solution with clear explanations and practice questions. Suitable for Grades 10-12.

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