(x-5i)(x%2B5i).jpeg "(x-2)(x-5i)(x+5i)")
Factoring and Expanding the Expression: (x-2)(x-5i)(x+5i)
This expression involves complex numbers and can be factored and expanded in a straightforward manner. Let's break it down step by step.
Understanding 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 the square root of -1.
Factoring the Expression
The given expression already appears partially factored:
- (x-2) is a simple linear factor.
- (x-5i) and (x+5i) are complex conjugates. Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts.
Expanding the Expression
Let's expand the expression to get a polynomial:
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Expand the complex conjugate pair: (x - 5i)(x + 5i) = x² - (5i)² = x² + 25 (Remember that i² = -1)
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Multiply the result by the remaining factor: (x² + 25)(x - 2) = x³ - 2x² + 25x - 50
Final Result
Therefore, the expanded form of the expression (x-2)(x-5i)(x+5i) is x³ - 2x² + 25x - 50.
Key Points
- Complex conjugates: The product of complex conjugates always results in a real number.
- Factoring with complex numbers: Factoring with complex numbers allows us to express polynomials in a more compact and insightful form.
- Understanding complex numbers is essential for advanced algebra and calculus concepts.
This example demonstrates a fundamental concept in complex number manipulation.