Question

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

\[
\boxed{x + 4}
\]