Given that f, of, x, equals, x, squared, minus, 13, x, plus, 36f(x)=x 
2
 −13x+36 and g, of, x, equals, x, minus, 4g(x)=x−4, find f, of, x, divided by, g, of, xf(x)÷g(x) and express the result as a polynomial in simplest form

Given the functions \( f(x) = x^2 - 13x + 36 \) and \( g(x) = x - 4 \), we want to find \( \frac{f(x)}{g(x)} \) and express it as a polynomial in simplest form.

### Step 1: Set up the division

We are asked to divide \( f(x) \) by \( g(x) \):
\[
\frac{f(x)}{g(x)} = \frac{x^2 - 13x + 36}{x - 4}
\]

### Step 2: Perform polynomial division

We divide \( x^2 - 13x + 36 \) by \( x - 4 \).

1. **Divide the leading term:**  
   \(\frac{x^2}{x} = x\)

2. **Multiply \( x \) by \( g(x) = x - 4 \):**  
   \( x \cdot (x - 4) = x^2 - 4x \)

3. **Subtract the result from \( f(x) \):**  
   \[
   (x^2 - 13x + 36) - (x^2 - 4x) = -9x + 36
   \]

4. **Repeat the process with the new polynomial:**  
   \(\frac{-9x}{x} = -9\)

5. **Multiply \(-9\) by \( g(x) \):**  
   \(-9 \cdot (x - 4) = -9x + 36\)

6. **Subtract the result:**  
   \[
   (-9x + 36) - (-9x + 36) = 0
   \]

Since there is no remainder, the division is complete.

### Final Result:
\[
\frac{f(x)}{g(x)} = x - 9
\]

Thus, the result of \( \frac{f(x)}{g(x)} \) expressed as a polynomial in simplest form is \( \boxed{x - 9} \).

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Would you like more details on any of the steps? 

Here are some related questions:
1. What happens when the remainder is not zero in polynomial division?
2. How do you factor \( f(x) = x^2 - 13x + 36 \) before dividing by \( g(x) \)?
3. What is synthetic division, and how does it compare to long division?
4. Can \( g(x) \) divide any other quadratic polynomials without remainder?
5. How do you solve \( f(x) = 0 \) and \( g(x) = 0 \) individually?

**Tip:** Polynomial division is similar to long division with numbers. Practice with different polynomials to get more comfortable with the steps.

Given that f(x) = x^2 - 13x + 36 and g(x) = x - 4, find f(x) ÷ g(x) and express the result as a polynomial in simplest form.

Learn how to perform polynomial division by dividing f(x) = x^2 - 13x + 36 by g(x) = x - 4 and express the result as a simplified polynomial. This problem covers key algebraic concepts and is suitable for students in Grades 9-10.

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