https://file.aiquickdraw.com/mathbot/file/17259405613021ef0rn97.pdf

The 4th problem in your document is as follows:

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**第4問 (配点: 20)**

Let \( p \) and \( q \) be real numbers, and define the complex number \( \alpha \) as \( \alpha = p + qi \).

Since \( \alpha - p = qi \), by squaring both sides, it follows that \( \alpha \) satisfies the equation:

\[
\alpha^2 - A\alpha + B = 0
\]

where \( A \) and \( B \) are specific expressions involving \( p \) and \( q \).

Next, when dividing \( x^3 \) by \( x^2 - A x + B \), the quotient \( Q(x) \) and remainder \( R(x) \) are given as:

\[
Q(x) = x + C, \quad R(x) = Dx - E
\]

where \( C \), \( D \), and \( E \) are expressions derived from \( p \) and \( q \).

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This is a summary of the fourth problem【6:13†source】.

Would you like more details or have any specific questions?

Here are five related questions:
1. How do you find the roots of a quadratic equation with complex numbers?
2. What is the geometric interpretation of a complex number in the plane?
3. How does the division of polynomials work in complex numbers?
4. What conditions must hold for a complex number \( \alpha^3 \) to be real?
5. Can the equation \( \alpha^2 - A\alpha + B = 0 \) be factored directly?

**Tip:** When squaring complex numbers, remember that \( i^2 = -1 \), which helps simplify expressions.

Explore how to solve a quadratic equation involving complex numbers and perform polynomial division. This math problem includes key concepts such as complex number manipulation and polynomial division. Suitable for advanced high school students in Grades 11-12.

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