(x^2-7x+11)/(x-4) the answer should give the quotient and the remainder

To find the quotient and remainder when dividing \( \frac{x^2 - 7x + 11}{x - 4} \), we use polynomial long division.

1. **Set up the division:**
   
   \[
   \frac{x^2 - 7x + 11}{x - 4}
   \]

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

3. **Multiply the entire denominator by this result and subtract from the numerator:**
   
   \[
   x \cdot (x - 4) = x^2 - 4x
   \]
   
   Subtract:
   
   \[
   (x^2 - 7x + 11) - (x^2 - 4x) = -3x + 11
   \]

4. **Repeat the process with the new polynomial:**
   
   \[
   \frac{-3x}{x} = -3
   \]

5. **Multiply and subtract again:**
   
   \[
   -3 \cdot (x - 4) = -3x + 12
   \]
   
   Subtract:
   
   \[
   (-3x + 11) - (-3x + 12) = -1
   \]

6. **Write down the quotient and the remainder:**
   
   The quotient is \( x - 3 \) and the remainder is \( -1 \).

Thus, the division yields:
\[ \frac{x^2 - 7x + 11}{x - 4} = x - 3 + \frac{-1}{x - 4} \]

### Summary
- **Quotient:** \( x - 3 \)
- **Remainder:** \( -1 \)

Would you like any additional details or have any questions?

Here are some related questions you might find useful:
1. How do you perform polynomial long division?
2. What is synthetic division and how does it compare to polynomial long division?
3. How can you check the result of a polynomial division?
4. What are the applications of polynomial division in calculus?
5. Can polynomial division be used to simplify complex rational expressions?
6. What is the significance of the remainder in polynomial division?
7. How do you handle polynomial division when the degrees of the polynomials are higher?
8. What happens when you divide by a polynomial of higher degree?

**Tip:** Always double-check your multiplication and subtraction steps to avoid simple errors in polynomial division.

Learn how to divide the polynomial x^2 - 7x + 11 by x - 4 using polynomial long division. Understand the quotient and remainder of this division with a detailed step-by-step solution. Suitable for students in Grades 11-12.

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