Use polynomial long division to rewrite the following fraction in the form q(x)+r(x)d(x), where d(x) is the denominator of the original fraction, q(x) is the quotient, and r(x) is the remainder.

9x4−18x3+6x2+3x+13x2+1

We are asked to perform polynomial long division on the following expression:

\[
\frac{9x^4 - 18x^3 + 6x^2 + 3x + 1}{13x^2 + 1}
\]

We will divide \(9x^4 - 18x^3 + 6x^2 + 3x + 1\) by \(13x^2 + 1\) using the polynomial long division process. Here's how it works step by step.

### Step 1: Divide the leading terms
- Divide the leading term of the numerator \(9x^4\) by the leading term of the denominator \(13x^2\):
  \[
  \frac{9x^4}{13x^2} = \frac{9}{13}x^2
  \]
  So, the first term of the quotient \(q(x)\) is \(\frac{9}{13}x^2\).

### Step 2: Multiply and subtract
- Multiply \(\frac{9}{13}x^2\) by the entire divisor \(13x^2 + 1\):
  \[
  \left(\frac{9}{13}x^2\right)(13x^2 + 1) = 9x^4 + \frac{9}{13}x^2
  \]
- Now, subtract this from the original polynomial:
  \[
  (9x^4 - 18x^3 + 6x^2 + 3x + 1) - (9x^4 + \frac{9}{13}x^2) = -18x^3 + \left(6x^2 - \frac{9}{13}x^2\right) + 3x + 1
  \]
  Simplify the expression:
  \[
  -18x^3 + \left(\frac{78}{13}x^2 - \frac{9}{13}x^2\right) + 3x + 1 = -18x^3 + \frac{69}{13}x^2 + 3x + 1
  \]

### Step 3: Divide the next leading terms
- Divide \(-18x^3\) by \(13x^2\):
  \[
  \frac{-18x^3}{13x^2} = \frac{-18}{13}x
  \]
  So, the second term of the quotient \(q(x)\) is \(\frac{-18}{13}x\).

### Step 4: Multiply and subtract
- Multiply \(\frac{-18}{13}x\) by the entire divisor \(13x^2 + 1\):
  \[
  \left(\frac{-18}{13}x\right)(13x^2 + 1) = -18x^3 + \frac{-18}{13}x
  \]
- Now, subtract this from the current remainder:
  \[
  (-18x^3 + \frac{69}{13}x^2 + 3x + 1) - (-18x^3 + \frac{-18}{13}x) = \frac{69}{13}x^2 + \left(3x - \frac{-18}{13}x\right) + 1
  \]
  Simplify the expression:
  \[
  \frac{69}{13}x^2 + \left(3x + \frac{18}{13}x\right) + 1 = \frac{69}{13}x^2 + \frac{57}{13}x + 1
  \]

### Step 5: Divide the next leading terms
- Divide \(\frac{69}{13}x^2\) by \(13x^2\):
  \[
  \frac{\frac{69}{13}x^2}{13x^2} = \frac{69}{169}
  \]
  So, the third term of the quotient \(q(x)\) is \(\frac{69}{169}\).

### Step 6: Multiply and subtract
- Multiply \(\frac{69}{169}\) by the entire divisor \(13x^2 + 1\):
  \[
  \left(\frac{69}{169}\right)(13x^2 + 1) = \frac{69}{13}x^2 + \frac{69}{169}
  \]
- Now, subtract this from the current remainder:
  \[
  \left(\frac{69}{13}x^2 + \frac{57}{13}x + 1\right) - \left(\frac{69}{13}x^2 + \frac{69}{169}\right) = \frac{57}{13}x + \left(1 - \frac{69}{169}\right)
  \]
  Simplify the expression:
  \[
  \frac{57}{13}x + \frac{169}{169} - \frac{69}{169} = \frac{57}{13}x + \frac{100}{169}
  \]

### Final Answer:
The quotient \(q(x)\) and remainder \(r(x)\) are:

\[
q(x) = \frac{9}{13}x^2 + \frac{-18}{13}x + \frac{69}{169}
\]
\[
r(x) = \frac{57}{13}x + \frac{100}{169}
\]

Thus, the fraction can be rewritten as:

\[
\frac{9x^4 - 18x^3 + 6x^2 + 3x + 1}{13x^2 + 1} = \left(\frac{9}{13}x^2 + \frac{-18}{13}x + \frac{69}{169}\right) + \frac{\frac{57}{13}x + \frac{100}{169}}{13x^2 + 1}
\]

Do you want more details, or have any questions?  
Here are 5 follow-up questions to expand on this:

1. How does polynomial long division differ from synthetic division?
2. What are some practical applications of polynomial long division in mathematics?
3. Can the remainder of a polynomial division ever be negative?
4. How does the degree of the remainder compare to the degree of the divisor?
5. What happens if the remainder is zero in polynomial long division?

**Tip:** When performing long division with polynomials, always focus on dividing the highest-degree terms first.

Use polynomial long division to rewrite the following fraction in the form q(x) + r(x)/d(x), where d(x) is the denominator of the original fraction, q(x) is the quotient, and r(x) is the remainder. 

(9x^4 − 18x^3 + 6x^2 + 3x + 1) / (13x^2 + 1)

Learn how to use polynomial long division to rewrite (9x^4 − 18x^3 + 6x^2 + 3x + 1) / (13x^2 + 1) into a quotient and remainder form. This guide provides a step-by-step solution, covering important algebraic concepts such as polynomial division and the division algorithm for polynomials. Ideal for Grades 10-12.

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