x⁴+3x³+6x-4÷x²+2 dividing polynomials long method 

To perform polynomial long division of \( \frac{x^4 + 3x^3 + 6x - 4}{x^2 + 2} \), we'll use the long division method step-by-step. 

### Step-by-Step Division

1. **First term division:**
   - Divide the leading term of the numerator (\(x^4\)) by the leading term of the denominator (\(x^2\)).
   \[
   \frac{x^4}{x^2} = x^2
   \]
   - This is the first term of the quotient.

2. **Multiply the denominator:**
   - Multiply \(x^2 + 2\) by \(x^2\):
   \[
   (x^2 + 2)(x^2) = x^4 + 2x^2
   \]
   - Subtract this result from the original polynomial:
   \[
   (x^4 + 3x^3 + 6x - 4) - (x^4 + 2x^2) = 3x^3 - 2x^2 + 6x - 4
   \]

3. **Second term division:**
   - Now divide the leading term of the new expression (\(3x^3\)) by the leading term of the denominator (\(x^2\)):
   \[
   \frac{3x^3}{x^2} = 3x
   \]
   - This is the next term of the quotient.

4. **Multiply the denominator:**
   - Multiply \(x^2 + 2\) by \(3x\):
   \[
   (x^2 + 2)(3x) = 3x^3 + 6x
   \]
   - Subtract this from the current remainder:
   \[
   (3x^3 - 2x^2 + 6x - 4) - (3x^3 + 6x) = -2x^2 - 4
   \]

5. **Third term division:**
   - Now divide the leading term of the new expression (\(-2x^2\)) by the leading term of the denominator (\(x^2\)):
   \[
   \frac{-2x^2}{x^2} = -2
   \]
   - This is the next term of the quotient.

6. **Multiply the denominator:**
   - Multiply \(x^2 + 2\) by \(-2\):
   \[
   (x^2 + 2)(-2) = -2x^2 - 4
   \]
   - Subtract this from the current remainder:
   \[
   (-2x^2 - 4) - (-2x^2 - 4) = 0
   \]

### Final Answer:

The quotient is:
\[
x^2 + 3x - 2
\]
The remainder is zero, so the division is exact.

Let me know if you'd like any details or further clarification!

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### Follow-up questions:
1. What is synthetic division, and how does it differ from long division?
2. Can polynomial division result in a non-zero remainder, and how do you represent it?
3. How do you factor polynomials using the division result?
4. How can polynomial long division help in solving rational equations?
5. How do you apply the Remainder Theorem using polynomial division?

**Tip:** Always ensure the dividend and divisor are written in standard form (descending powers of \(x\)) before starting the division process.

x⁴ + 3x³ + 6x - 4 ÷ x² + 2 dividing polynomials long method

Learn how to perform polynomial long division for x⁴ + 3x³ + 6x - 4 ÷ x² + 2. This step-by-step guide covers key algebraic concepts, including polynomial division and the remainder theorem. Ideal for high school students in Grades 10-12.

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