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:

$$\boxed{{\displaystyle x+4}}$$