Given that f, of, x, equals, x, squared, minus, 5, x, minus, 36f(x)=x

Question

2
−5x−36 and g, of, x, equals, x, minus, 9g(x)=x−9, find f, of, x, divided by, g, of, xf(x)÷g(x) and express the result as a polynomial in simplest form.

Answer 1

To find \( \frac{f(x)}{g(x)} \) where \( f(x) = x^2 - 5x - 36 \) and \( g(x) = x - 9 \), we can directly substitute these functions into the expression:

\[ \frac{f(x)}{g(x)} = \frac{x^2 - 5x - 36}{x - 9} \]

Next, we need to divide \( f(x) \) by \( g(x) \). This can be done using polynomial long division.

1. Long Division Setup:

   Divide \( x^2 - 5x - 36 \) by \( x - 9 \).

2. Divide the leading term:

   The leading term of \( x^2 - 5x - 36 \) is \( x^2 \), and the leading term of \( x - 9 \) is \( x \). \[ \frac{x^2}{x} = x \]

3. Multiply and Subtract:

   Multiply \( x \) by \( x - 9 \) and subtract: \[ x(x - 9) = x^2 - 9x \]

   Now, subtract: \[ (x^2 - 5x - 36) - (x^2 - 9x) = -5x + 9x - 36 = 4x - 36 \]

4. Repeat the Process:

   Now divide \( 4x - 36 \) by \( x - 9 \): \[ \frac{4x}{x} = 4 \]

   Multiply \( 4 \) by \( x - 9 \): \[ 4(x - 9) = 4x - 36 \]

   Subtract: \[ (4x - 36) - (4x - 36) = 0 \]

5. Conclusion of the Division:

   The result of the division is: \[ x + 4 \]

So, we have: \[ \frac{f(x)}{g(x)} = x + 4 \]

Thus, expressing the result as a polynomial in simplest form, we find:

\[ \boxed{x + 4} \]