Find 
f(x)
g(x)
 = 

f
g

(x).
 Simplify using long division.
f(x) = 9x2 − 18x − 7    g(x) = 3x + 1

f
g

(x) = 


Given the functions:

\[
f(x) = 9x^2 - 18x - 7
\]
\[
g(x) = 3x + 1
\]

We need to find \( \frac{f(x)}{g(x)} \) and simplify it using long division.

### Step 1: Set up the long division
We divide \( 9x^2 - 18x - 7 \) by \( 3x + 1 \).

### Step 2: Perform the division
1. Divide the first term of the numerator by the first term of the denominator:  
   \[
   \frac{9x^2}{3x} = 3x
   \]
2. Multiply \( 3x \) by the entire divisor \( 3x + 1 \):  
   \[
   3x \cdot (3x + 1) = 9x^2 + 3x
   \]
3. Subtract this from the original polynomial:
   \[
   (9x^2 - 18x - 7) - (9x^2 + 3x) = -21x - 7
   \]
4. Divide the first term of the new polynomial by the first term of the divisor:
   \[
   \frac{-21x}{3x} = -7
   \]
5. Multiply \( -7 \) by the entire divisor \( 3x + 1 \):
   \[
   -7 \cdot (3x + 1) = -21x - 7
   \]
6. Subtract this result from the previous polynomial:
   \[
   (-21x - 7) - (-21x - 7) = 0
   \]

### Step 3: Write the result
Since the remainder is 0, the quotient is the simplified form of \( \frac{f(x)}{g(x)} \).

Thus:
\[
\frac{f(x)}{g(x)} = 3x - 7
\]

### Final Answer:
\[
\frac{f(x)}{g(x)} = 3x - 7
\]

Would you like more details on this process, or do you have any questions?

Here are some related questions for further exploration:
1. How do you perform polynomial long division with a remainder?
2. What is synthetic division, and how does it differ from long division?
3. How do you find the roots of a quadratic function like \( f(x) = 9x^2 - 18x - 7 \)?
4. What happens if the degree of the numerator is less than the degree of the denominator?
5. How can polynomial division be applied in calculus, for example, in finding limits?

**Tip:** When performing long division of polynomials, always align terms with the same degree under each other for clarity and accuracy.

Learn how to simplify polynomial division using long division. Step-by-step solution for dividing f(x) = 9x^2 - 18x - 7 by g(x) = 3x + 1, resulting in f/g(x) = 3x - 7. Suitable for high school students.

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Find f(x) g(x)

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