(x-3i)(x%2B3i).jpeg "(x-5)(x-3i)(x+3i)")
Factoring and Expanding Complex Polynomials: An Example
This article will explore the expansion of the polynomial: (x - 5)(x - 3i)(x + 3i).
Understanding Complex Numbers
Before delving into the expansion, let's briefly discuss complex numbers. Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit defined as √-1.
Expanding the Polynomial
To expand the given polynomial, we will use the distributive property and simplify. Here's the breakdown:
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Focus on the complex conjugates: Notice that (x - 3i) and (x + 3i) are complex conjugates. This means they only differ in the sign of the imaginary part.
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Multiply the complex conjugates: (x - 3i)(x + 3i) = x² - (3i)² = x² + 9
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Multiply the result by (x - 5): (x² + 9)(x - 5) = x³ - 5x² + 9x - 45
The Final Result
Therefore, the expanded form of (x - 5)(x - 3i)(x + 3i) is x³ - 5x² + 9x - 45.
Key Takeaways
- Complex conjugates simplify: Multiplying complex conjugates eliminates the imaginary term, resulting in a real number.
- Distributive property is key: Expanding polynomials often involves applying the distributive property multiple times.
- Resulting polynomial is real: Even though the original expression involves complex numbers, the expanded form is a real-valued polynomial.
This example demonstrates how complex numbers can be incorporated into polynomial expressions and how their unique properties can be used for simplification.