3x⁴-x³+x-2 ÷ 3x²+1/3 in synthetic division

### Synthetic Division of \( \frac{3x^4 - x^3 + x - 2}{3x^2 + \frac{1}{3}} \)

Synthetic division is typically used when dividing a polynomial by a **linear divisor** like \(x - c\). In this case, the divisor \(3x^2 + \frac{1}{3}\) is a quadratic, so synthetic division isn't directly applicable in its basic form.

However, if we modify the divisor to something synthetic division can handle, the process becomes clearer. Let’s consider the modified divisor:

\[
3x^2 + \frac{1}{3} = \frac{9x^2 + 1}{3}
\]

To make the division easier to manage, multiply the entire numerator by 3 so that we’re dividing by \(9x^2 + 1\) instead:

\[
\frac{3(3x^4 - x^3 + x - 2)}{9x^2 + 1} = \frac{9x^4 - 3x^3 + 3x - 6}{9x^2 + 1}
\]

Now, we are left with dividing by a quadratic, which synthetic division is not built for.

Would you like to proceed with polynomial long division for this specific case, or would you like help with another problem?

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Here are 5 questions you could explore next:
1. How is synthetic division adapted for different types of divisors?
2. What methods exist for dividing polynomials by quadratic expressions?
3. Why does synthetic division work for linear but not higher-degree divisors?
4. How can factoring help simplify division of polynomials with higher-degree divisors?
5. Can polynomial division be used in calculus for solving limit problems?

**Tip:** For quadratic divisors, long division is generally the most reliable method.

Explore how to divide the polynomial 3x⁴ - x³ + x - 2 by 3x² + 1/3 using polynomial division techniques. This problem involves higher-degree polynomials and requires understanding synthetic division concepts. Suitable for advanced high school students and college-level learners.

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