(x-4i)(x%2B4i).jpeg "(x-2)(x-4i)(x+4i)")
Multiplying Complex Numbers: (x-2)(x-4i)(x+4i)
This problem involves multiplying three factors, one of which is a real binomial and the other two are complex conjugates. Let's break down the steps to solve it.
Understanding Complex Conjugates
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. For example, the complex conjugate of (a + bi) is (a - bi). A crucial property of complex conjugates is that their product always results in a real number.
Solving the Problem
-
Focus on the complex conjugates: Start by multiplying (x - 4i) and (x + 4i):
(x - 4i)(x + 4i) = x² + 4xi - 4xi - 16i²
Since i² = -1, we can simplify:
x² + 4xi - 4xi - 16i² = x² + 16
-
Multiply the result by the real binomial: Now, multiply the result (x² + 16) by (x - 2):
(x² + 16)(x - 2) = x³ - 2x² + 16x - 32
Conclusion
Therefore, the expanded form of (x - 2)(x - 4i)(x + 4i) is x³ - 2x² + 16x - 32. This demonstrates how the product of complex conjugates simplifies the expression, ultimately leading to a real polynomial.