(x-2i)(x%2B2i).jpeg "(x-3)(x-2i)(x+2i)")
Exploring the Polynomial (x-3)(x-2i)(x+2i)
This article will delve into the polynomial (x-3)(x-2i)(x+2i), exploring its properties and how to work with it.
Understanding the Factors
- (x-3): This factor represents a linear term, indicating a root of the polynomial at x = 3.
- (x-2i): This factor is a complex term, where i is the imaginary unit (√-1). This indicates a complex root at x = 2i.
- (x+2i): This factor is also complex, indicating another complex root at x = -2i.
Notice that the complex roots appear as a conjugate pair. This is a common feature in polynomials with real coefficients, where complex roots always come in pairs.
Expanding the Polynomial
To understand the full form of the polynomial, we need to expand the product:
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Start with the complex conjugate pair: (x-2i)(x+2i) = x² - (2i)² = x² + 4
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Multiply the result by the linear term: (x² + 4)(x-3) = x³ - 3x² + 4x - 12
Therefore, the expanded form of the polynomial is x³ - 3x² + 4x - 12.
Key Properties
- Degree: The highest power of x in the polynomial is 3, making it a cubic polynomial.
- Real Coefficients: All coefficients in the polynomial are real numbers.
- Roots: The polynomial has three roots:
- x = 3 (real root)
- x = 2i (complex root)
- x = -2i (complex root)
Significance
This polynomial provides a basic example of how complex roots arise in polynomial equations and how they always appear in conjugate pairs. It demonstrates the relationship between the roots of a polynomial and its factored form.
By understanding the factors and expansion of this polynomial, we gain insights into the nature of polynomials with complex roots and how they can be manipulated and analyzed.