Long division for A⁸-5A⁷+7A⁶-3A⁵+A⁴-5A³+8A²-2A+I by A³-5A²+7A-3I

To divide the polynomial \( A^8 - 5A^7 + 7A^6 - 3A^5 + A^4 - 5A^3 + 8A^2 - 2A + I \) by \( A^3 - 5A^2 + 7A - 3I \), we will perform polynomial long division. Let's go through the steps:

### Step-by-Step Long Division:

1. **Divide the first term** of the dividend \( A^8 \) by the first term of the divisor \( A^3 \):
   \[
   \frac{A^8}{A^3} = A^5
   \]
   This is the first term of the quotient.

2. **Multiply** the entire divisor by \( A^5 \) and subtract the result from the original polynomial:
   \[
   A^5 \cdot (A^3 - 5A^2 + 7A - 3I) = A^8 - 5A^7 + 7A^6 - 3A^5
   \]
   Subtracting this from the original polynomial:
   \[
   (A^8 - 5A^7 + 7A^6 - 3A^5 + A^4 - 5A^3 + 8A^2 - 2A + I) - (A^8 - 5A^7 + 7A^6 - 3A^5)
   \]
   The result is:
   \[
   A^4 - 5A^3 + 8A^2 - 2A + I
   \]

3. **Divide the first term** of the new dividend \( A^4 \) by the first term of the divisor \( A^3 \):
   \[
   \frac{A^4}{A^3} = A
   \]
   This is the second term of the quotient.

4. **Multiply** the entire divisor by \( A \) and subtract the result from the new dividend:
   \[
   A \cdot (A^3 - 5A^2 + 7A - 3I) = A^4 - 5A^3 + 7A^2 - 3A
   \]
   Subtracting this from the new dividend:
   \[
   (A^4 - 5A^3 + 8A^2 - 2A + I) - (A^4 - 5A^3 + 7A^2 - 3A)
   \]
   The result is:
   \[
   A^2 + A + I
   \]

5. **Divide the first term** of the new dividend \( A^2 \) by the first term of the divisor \( A^3 \):
   \[
   \frac{A^2}{A^3} = 0
   \]
   So we can't divide further, and \( A^2 + A + I \) is the remainder.

### Final Answer:
The quotient is:
\[
A^5 + A
\]
And the remainder is:
\[
A^2 + A + I
\]

Thus, the result of the division is:
\[
A^5 + A + \frac{A^2 + A + I}{A^3 - 5A^2 + 7A - 3I}
\]

Would you like more details or have any questions?

### 5 Related Questions:
1. How is synthetic division used for polynomials?
2. What is the remainder theorem in polynomial division?
3. How does polynomial long division differ from regular numerical division?
4. Can we use the factor theorem to simplify polynomial division?
5. What is the relationship between the degree of the divisor and the quotient in polynomial division?

### Tip:
Always ensure that the degrees of the dividend and divisor align before starting polynomial long division to simplify the process.

Learn how to perform polynomial long division on the expression A⁸-5A⁷+7A⁶-3A⁵+A⁴-5A³+8A²-2A+I divided by A³-5A²+7A-3I. This detailed, step-by-step solution covers key algebraic concepts suitable for advanced high school and college students.

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