divide polynomial n(x) = x^5 + x^4 + 2 with a polynomial x +1 using the dividing angle

We are tasked with dividing the polynomial \( n(x) = x^5 + x^4 + 2 \) by \( d(x) = x + 1 \) using **long division** (also called the dividing angle method).

### Step-by-Step Long Division

We follow the steps of polynomial division:

1. **Set up the division**:
   - We are dividing \( x^5 + x^4 + 0x^3 + 0x^2 + 0x + 2 \) (fill in the missing powers of \( x \)) by \( x + 1 \).
   
2. **First division step**: Divide the leading term of \( n(x) \) by the leading term of \( d(x) \).
   - \( \frac{x^5}{x} = x^4 \)

3. **Multiply** \( x^4 \) by \( x + 1 \) and subtract:
   - \( x^4(x + 1) = x^5 + x^4 \)
   - Subtract: \( (x^5 + x^4) - (x^5 + x^4) = 0 \)

4. **Bring down the next term** (which is \( 0x^3 \)):
   - The remainder is \( 0x^3 + 0x^2 + 0x + 2 \).

5. **Next division step**: Divide the leading term of the new remainder by \( x \).
   - \( \frac{0x^3}{x} = 0 \)
   - Multiply and subtract: Nothing changes, so the remainder is still \( 0x^3 + 0x^2 + 0x + 2 \).

6. **Repeat this for the rest** of the terms:
   - Since there are no more higher powers of \( x \), we proceed to handle the constant term.
   - The final remainder is \( 2 \).

Thus, the quotient is \( x^4 \) and the remainder is \( 2 \).

### Final Answer:
\[
\frac{x^5 + x^4 + 2}{x + 1} = x^4 \quad \text{with remainder} \quad 2
\]

Would you like more details or have any questions?

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Here are some related questions for further exploration:

1. How do you divide higher-degree polynomials using synthetic division?
2. What happens when the divisor has more terms (e.g., \(x^2 + 2x + 1\))?
3. Can long division be used with non-monic divisors (where the leading coefficient isn't 1)?
4. How does polynomial division relate to factorization?
5. How can you check if a remainder is correct in a polynomial division?

**Tip**: The Remainder Theorem states that when a polynomial is divided by \( x - a \), the remainder is the value of the polynomial evaluated at \( a \).

Explore how to divide the polynomial x^5 + x^4 + 2 by x + 1 using the long division method. This step-by-step guide covers polynomial division, quotient calculation, and remainder analysis, incorporating the Remainder Theorem. Ideal for students in Grades 10-12.

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