Use long division to find the quotient and the remainder.

StartFraction 5 x Superscript 4 Baseline minus 6 x squared plus 2 x minus 2 Over x plus 2 EndFraction5x4−6x2+2x−2x+2

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Part 1

The quotient is 

enter your response here 

and the remainder is 

enter your response here.




To solve the division of the polynomial \( \frac{5x^4 - 6x^2 + 2x - 2}{x + 2} \) using long division, follow these steps:

### Step-by-Step Long Division:

1. **Divide the first term:**
   - Divide \( 5x^4 \) by \( x \). The result is \( 5x^3 \).

2. **Multiply the divisor by the result:**
   - Multiply \( 5x^3 \) by \( x + 2 \), which gives \( 5x^4 + 10x^3 \).

3. **Subtract:**
   \[
   (5x^4 - 6x^2 + 2x - 2) - (5x^4 + 10x^3) = -10x^3 - 6x^2 + 2x - 2
   \]

4. **Divide the new first term:**
   - Divide \( -10x^3 \) by \( x \), which gives \( -10x^2 \).

5. **Multiply the divisor by this result:**
   - Multiply \( -10x^2 \) by \( x + 2 \), which gives \( -10x^3 - 20x^2 \).

6. **Subtract:**
   \[
   (-10x^3 - 6x^2 + 2x - 2) - (-10x^3 - 20x^2) = 14x^2 + 2x - 2
   \]

7. **Divide the new first term:**
   - Divide \( 14x^2 \) by \( x \), which gives \( 14x \).

8. **Multiply the divisor by this result:**
   - Multiply \( 14x \) by \( x + 2 \), which gives \( 14x^2 + 28x \).

9. **Subtract:**
   \[
   (14x^2 + 2x - 2) - (14x^2 + 28x) = -26x - 2
   \]

10. **Divide the new first term:**
    - Divide \( -26x \) by \( x \), which gives \( -26 \).

11. **Multiply the divisor by this result:**
    - Multiply \( -26 \) by \( x + 2 \), which gives \( -26x - 52 \).

12. **Subtract:**
    \[
    (-26x - 2) - (-26x - 52) = 50
    \]

Thus, the quotient is \( 5x^3 - 10x^2 + 14x - 26 \), and the remainder is 50.

### Final Answer:
- **Quotient:** \( 5x^3 - 10x^2 + 14x - 26 \)
- **Remainder:** 50

Would you like any more details or have any questions?

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Here are 5 related questions:
1. How do you verify the result of polynomial long division?
2. What is synthetic division, and when is it used?
3. Can you explain how to handle remainders in polynomial division?
4. How does polynomial long division compare to numerical long division?
5. How would you handle division by a polynomial with a degree greater than 1?

**Tip:** Always check your division by multiplying the quotient by the divisor and adding the remainder; this should give the original polynomial.

Use long division to find the quotient and the remainder. \( \frac{5x^4 - 6x^2 + 2x - 2}{x + 2} \)

Learn how to use long division to divide the polynomial 5x^4 - 6x^2 + 2x - 2 by x + 2 and find the quotient and remainder. This step-by-step solution demonstrates the application of the long division algorithm for polynomials, suitable for high school students in Grades 9-12.

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