(x-3i)(x%2B3i).jpeg "(x-2)(x-3i)(x+3i)")
Exploring the Polynomial (x-2)(x-3i)(x+3i)
This article delves into the polynomial (x-2)(x-3i)(x+3i) and explores its properties, including its roots, expansion, and relationship to the complex number system.
Understanding the Roots
The polynomial is given in factored form, which directly reveals its roots. A root of a polynomial is a value of x that makes the polynomial equal to zero.
- x = 2 Setting (x-2) equal to zero, we find x = 2 is a root.
- x = 3i Setting (x-3i) equal to zero, we find x = 3i is a root.
- x = -3i Setting (x+3i) equal to zero, we find x = -3i is a root.
Expanding the Polynomial
To better understand the polynomial's structure, we can expand it:
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Focus on the Complex Factors: (x - 3i)(x + 3i) = x² - (3i)² = x² + 9
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Multiply with the Remaining Factor: (x² + 9)(x - 2) = x³ - 2x² + 9x - 18
Therefore, the expanded form of the polynomial is x³ - 2x² + 9x - 18.
Complex Conjugate Pairs
Notice that the roots 3i and -3i are complex conjugates. This is a common occurrence in polynomials with real coefficients. Complex roots always appear in conjugate pairs. This ensures that the polynomial's coefficients remain real.
Conclusion
The polynomial (x-2)(x-3i)(x+3i) is a cubic polynomial with one real root (x = 2) and two complex roots (x = 3i and x = -3i). Understanding its factored form allows us to quickly identify the roots, while expanding the polynomial reveals its standard form. The presence of complex conjugate pairs highlights a key property of polynomials with real coefficients.