3x⁴-x³+x-2 ÷ 3x²+x+1 division of polynomial and synthetic division

### Polynomial Division (Long Division)

We will first solve the problem using polynomial long division. We are dividing:

\[
\frac{3x^4 - x^3 + x - 2}{3x^2 + x + 1}
\]

**Step 1:** Divide the leading term of the dividend \(3x^4\) by the leading term of the divisor \(3x^2\).

\[
\frac{3x^4}{3x^2} = x^2
\]

Now, multiply the entire divisor by \(x^2\):

\[
x^2(3x^2 + x + 1) = 3x^4 + x^3 + x^2
\]

Subtract this from the original polynomial:

\[
(3x^4 - x^3 + x - 2) - (3x^4 + x^3 + x^2) = -2x^3 - x^2 + x - 2
\]

**Step 2:** Divide the leading term \(-2x^3\) by the leading term \(3x^2\).

\[
\frac{-2x^3}{3x^2} = -\frac{2}{3}x
\]

Now, multiply the entire divisor by \(-\frac{2}{3}x\):

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

Subtract this from the current remainder:

\[
(-2x^3 - x^2 + x - 2) - (-2x^3 - \frac{2}{3}x^2 - \frac{2}{3}x) = \frac{-1}{3}x^2 + \frac{5}{3}x - 2
\]

**Step 3:** Divide the leading term \(\frac{-1}{3}x^2\) by the leading term \(3x^2\).

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

Now, multiply the entire divisor by \(-\frac{1}{9}\):

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

Subtract this from the current remainder:

\[
\left(\frac{-1}{3}x^2 + \frac{5}{3}x - 2\right) - \left(-\frac{1}{3}x^2 - \frac{1}{9}x - \frac{1}{9}\right) = \frac{16}{9}x - \frac{17}{9}
\]

Thus, the quotient is:

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

and the remainder is:

\[
\frac{16}{9}x - \frac{17}{9}
\]

So, the result of the division is:

\[
\frac{3x^4 - x^3 + x - 2}{3x^2 + x + 1} = x^2 - \frac{2}{3}x - \frac{1}{9} + \frac{\frac{16}{9}x - \frac{17}{9}}{3x^2 + x + 1}
\]

### Synthetic Division
Since the divisor is of degree 2, synthetic division is not possible here (synthetic division works only when dividing by linear divisors like \(x - c\)).

Would you like more details about any specific step, or have any questions?

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Here are 5 related questions you can explore:
1. How does synthetic division work for linear divisors?
2. What is the difference between long division and synthetic division?
3. Can you perform division of polynomials with higher-degree divisors using synthetic division?
4. How do you solve polynomial remainder problems?
5. How can you use the remainder to check the correctness of polynomial division?

**Tip:** When performing polynomial long division, always align terms by degree to avoid mistakes during subtraction.

Learn how to divide polynomials using polynomial long division with the problem 3x⁴ - x³ + x - 2 ÷ 3x² + x + 1. Understand the step-by-step process and get insights into polynomial division techniques. Suitable for advanced high school students and college-level learners.

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