(x-4i)(x%2B4i).jpeg "(x-4)(x-4i)(x+4i)")
Factoring and Solving the Polynomial (x-4)(x-4i)(x+4i)
This expression represents a polynomial in factored form. Let's break down what this means and how to solve it.
Understanding the Factors
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(x-4): This is a simple linear factor, representing a root of the polynomial at x = 4.
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(x-4i): This factor introduces a complex root at x = 4i, where 'i' is the imaginary unit (√-1).
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(x+4i): This is the conjugate of the previous factor. It also represents a complex root at x = -4i.
Expanding the Polynomial
To understand the polynomial's behavior, we can expand it by multiplying the factors:
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Start with the complex conjugate factors: (x - 4i)(x + 4i) = x² - (4i)² = x² + 16
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Multiply the result by the remaining factor: (x² + 16)(x - 4) = x³ - 4x² + 16x - 64
Therefore, the expanded form of the polynomial is x³ - 4x² + 16x - 64.
Key Points
- Real Root: The polynomial has one real root at x = 4.
- Complex Roots: The polynomial has two complex roots at x = 4i and x = -4i.
- Conjugate Pairs: Complex roots always appear in conjugate pairs. This is because the coefficients of the polynomial are real.
- Symmetry: The graph of this polynomial will exhibit symmetry about the real axis due to the complex conjugate roots.
Conclusion
The expression (x-4)(x-4i)(x+4i) represents a cubic polynomial with one real root and two complex conjugate roots. By expanding the factored form, we can understand the polynomial's behavior and its relationship to the complex number system.