Divide u^3 by u+1 using polynomial division method 

To divide \( u^3 \) by \( u + 1 \) using the polynomial division method, follow these steps:

### Step 1: Set Up the Division
We want to divide \( u^3 \) by \( u + 1 \).

So, write it as:
\[
\frac{u^3}{u+1}
\]

### Step 2: Divide the Leading Term
Divide the leading term of the dividend (\( u^3 \)) by the leading term of the divisor (\( u \)):

\[
\frac{u^3}{u} = u^2
\]

Place \( u^2 \) above the division line.

### Step 3: Multiply and Subtract
Multiply \( u^2 \) by the entire divisor (\( u + 1 \)):

\[
u^2 \times (u + 1) = u^3 + u^2
\]

Subtract this result from the original polynomial:

\[
(u^3 - (u^3 + u^2)) = -u^2
\]

### Step 4: Repeat the Process
Now, divide the new leading term (\(-u^2\)) by the leading term of the divisor (\( u \)):

\[
\frac{-u^2}{u} = -u
\]

Place \(-u\) above the division line next to \( u^2 \).

Multiply \(-u\) by the entire divisor:

\[
-u \times (u + 1) = -u^2 - u
\]

Subtract this from the current remainder:

\[
(-u^2 - (-u^2 - u)) = u
\]

### Step 5: Divide Again
Next, divide the new leading term (\( u \)) by the leading term of the divisor (\( u \)):

\[
\frac{u}{u} = 1
\]

Place \( 1 \) above the division line next to \(-u\).

Multiply \( 1 \) by the entire divisor:

\[
1 \times (u + 1) = u + 1
\]

Subtract this from the current remainder:

\[
(u - (u + 1)) = -1
\]

### Step 6: Combine Results
The final quotient is \( u^2 - u + 1 \), and the remainder is \(-1\).

So, the result of the division is:

\[
\frac{u^3}{u+1} = u^2 - u + 1 - \frac{1}{u+1}
\]

### Final Answer:
\[
\text{Quotient: } u^2 - u + 1, \quad \text{Remainder: } -1
\]

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you perform polynomial long division when the divisor is more complex?
2. Can the remainder in polynomial division ever be a higher degree than the divisor?
3. How does synthetic division differ from long division for polynomials?
4. What happens when you divide a polynomial by a binomial where the leading coefficient isn't 1?
5. How can you verify the result of a polynomial division?

**Tip:** Always check your remainder's degree; it should be lower than the degree of the divisor.

Learn how to divide u^3 by u+1 using polynomial division method with a step-by-step solution. This problem covers polynomial division concepts and the quotient and remainder theorem. Suitable for advanced high school students.

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