Divide $$ x^4-6x^3+2x^2-x-9\text{ by x}+1 $$

To divide the polynomial $ x^4 - 6x^3 + 2x^2 - x - 9 $ by $ x + 1 $, we'll use polynomial long division.

### Step 1: Set up the division

We want to divide $ x^4 - 6x^3 + 2x^2 - x - 9 $ by $ x + 1 $.

### Step 2: Divide the first term
Divide the leading term of the dividend $ x^4 $ by the leading term of the divisor $ x $:
\[
\frac{x^4}{x} = x^3
\]
This is the first term of the quotient.

### Step 3: Multiply and subtract
Multiply $ x^3 $ by $ x + 1 $ and subtract the result from the original polynomial:
\[
x^3(x + 1) = x^4 + x^3
\]
Subtract this from the original polynomial:
\[
(x^4 - 6x^3 + 2x^2 - x - 9) - (x^4 + x^3) = -7x^3 + 2x^2 - x - 9
\]

### Step 4: Repeat the process
Now divide the leading term $ -7x^3 $ by $ x $:
\[
\frac{-7x^3}{x} = -7x^2
\]
Multiply $ -7x^2 $ by $ x + 1 $ and subtract:
\[
-7x^2(x + 1) = -7x^3 - 7x^2
\]
Subtract this from the current polynomial:
\[
(-7x^3 + 2x^2 - x - 9) - (-7x^3 - 7x^2) = 9x^2 - x - 9
\]

### Step 5: Continue dividing
Divide the leading term $ 9x^2 $ by $ x $:
\[
\frac{9x^2}{x} = 9x
\]
Multiply $ 9x $ by $ x + 1 $ and subtract:
\[
9x(x + 1) = 9x^2 + 9x
\]
Subtract this from the current polynomial:
\[
(9x^2 - x - 9) - (9x^2 + 9x) = -10x - 9
\]

### Step 6: Final division
Divide the leading term $ -10x $ by $ x $:
\[
\frac{-10x}{x} = -10
\]
Multiply $ -10 $ by $ x + 1 $ and subtract:
\[
-10(x + 1) = -10x - 10
\]
Subtract this from the current polynomial:
\[
(-10x - 9) - (-10x - 10) = 1
\]

### Conclusion
The quotient is $ x^3 - 7x^2 + 9x - 10 $ and the remainder is $ 1 $.

So, 
\[
\frac{x^4 - 6x^3 + 2x^2 - x - 9}{x + 1} = x^3 - 7x^2 + 9x - 10 + \frac{1}{x + 1}
\]

Would you like more details or have any questions?

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Here are five related questions you might consider:

1. How would you perform synthetic division for the same polynomial division problem?
2. What is the significance of the remainder when dividing polynomials?
3. How can you verify the result of polynomial division?
4. How does polynomial division relate to factoring polynomials?
5. What happens if you divide $ x^4 - 6x^3 + 2x^2 - x - 9 $ by a different binomial, like $ x - 2 $?

**Tip:** When dividing polynomials, always arrange the terms in descending order of the variable's power to avoid mistakes.

Learn how to divide the polynomial x^4 - 6x^3 + 2x^2 - x - 9 by x + 1 using polynomial long division. This problem involves step-by-step division of polynomials, understanding the quotient and remainder. Suitable for high school students studying polynomial division.

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