divide (3x⁴ - x³+ x-2) ÷ (3x² + x+1)

To divide \( (3x^4 - x^3 + x - 2) \) by \( (3x^2 + x + 1) \), we'll use polynomial long division. Here's the process step by step:

### Step 1: Set up the division
We will divide \( 3x^4 - x^3 + 0x^2 + x - 2 \) by \( 3x^2 + x + 1 \).

### Step 2: Divide the leading terms
The first term of the quotient comes from dividing the leading term of the dividend by the leading term of the divisor:

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

So, the first term of the quotient is \( x^2 \).

### Step 3: Multiply and subtract
Multiply \( x^2 \) by \( (3x^2 + x + 1) \):

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

Now subtract this from the original polynomial:

\[
(3x^4 - x^3 + 0x^2 + x - 2) - (3x^4 + x^3 + x^2) = -2x^3 - x^2 + x - 2
\]

### Step 4: Divide the next leading term
Now, divide the leading term of the new polynomial by the leading term of the divisor:

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

So, the next term in the quotient is \( -\frac{2}{3}x \).

### Step 5: Multiply and subtract
Multiply \( -\frac{2}{3}x \) by \( (3x^2 + x + 1) \):

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

Subtract this from the current polynomial:

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

### Step 6: Divide the next leading term
Now, divide the leading term of the new polynomial by the leading term of the divisor:

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

So, the next term in the quotient is \( -\frac{1}{9} \).

### Step 7: Multiply and subtract
Multiply \( -\frac{1}{9} \) by \( (3x^2 + x + 1) \):

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

Subtract this from the current polynomial:

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

### Step 8: Write the quotient and remainder
The quotient is:

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

The remainder is:

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

Thus, the division can be written as:

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

---

Would you like more details or clarification on any step? 

Here are 5 related questions to explore further:
1. How do you perform synthetic division on polynomials?
2. What is the Remainder Theorem in polynomial division?
3. How do you solve a polynomial equation after dividing?
4. Can you use polynomial division to factor expressions?
5. How does polynomial long division relate to rational functions?

**Tip:** Polynomial long division follows a similar process to regular long division of numbers. Keep track of terms carefully for success!

divide (3x⁴ - x³ + x - 2) ÷ (3x² + x + 1)

Learn how to divide polynomials using long division with the example (3x⁴ - x³ + x - 2) ÷ (3x² + x + 1). Follow step-by-step guidance through polynomial division, including detailed explanations of each calculation. Perfect for students in Grades 10-12 studying algebra and rational expressions.

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