(x-5i)(x%2B5i).jpeg "(x-4)(x-5i)(x+5i)")
Factoring and Expanding Complex Polynomials: (x-4)(x-5i)(x+5i)
This article will explore the process of factoring and expanding the polynomial expression (x-4)(x-5i)(x+5i). We will delve into the concepts of complex numbers and their role in polynomial factorization.
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 √(-1). The term bi is called the imaginary part.
Factoring the Polynomial
The given polynomial (x-4)(x-5i)(x+5i) is already in factored form. Notice that the last two factors represent a difference of squares: (x-5i)(x+5i).
Expanding the Polynomial
To expand the polynomial, we can use the distributive property (FOIL method) and the fact that i² = -1:
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Expand the difference of squares: (x-5i)(x+5i) = x² - (5i)² = x² - 25i² = x² + 25
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Multiply the result by (x-4): (x-4)(x² + 25) = x³ + 25x - 4x² - 100
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Combine like terms: x³ - 4x² + 25x - 100
Therefore, the expanded form of the polynomial (x-4)(x-5i)(x+5i) is x³ - 4x² + 25x - 100.
Significance of Complex Numbers
This example demonstrates the importance of complex numbers in polynomial factorization. Even though the original polynomial appears to have only real coefficients, the factorization involves complex factors. This highlights that complex numbers are essential for understanding the full structure of polynomials, even those with seemingly real coefficients.
Conclusion
By understanding the concepts of complex numbers and applying the distributive property, we can successfully factor and expand the polynomial (x-4)(x-5i)(x+5i). This process reveals the intricate relationship between real and complex numbers in polynomial algebra.