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Multiplying Complex Conjugates: (x-1-4i)(x-1+4i)
This expression involves multiplying two complex numbers which are complex conjugates. Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts.
Understanding Complex Conjugates
- Definition: If a complex number is of the form a + bi, its conjugate is a - bi.
- Key Property: The product of a complex number and its conjugate is always a real number. This is because the imaginary terms cancel out.
The Multiplication
Let's multiply the given expression:
(x-1-4i)(x-1+4i)
We can use the FOIL method (First, Outer, Inner, Last) to expand the product:
- First: (x * x) = x²
- Outer: (x * 4i) = 4ix
- Inner: (-1 * -1) = 1
- Last: (-1 * 4i) = -4i
- Combining like terms: x² + 4ix + 1 - 4i
Notice that the terms containing i cancel each other out:
- Simplified expression: x² + 1
Conclusion
As expected, the product of the complex conjugates (x-1-4i) and (x-1+4i) is a real number: x² + 1. This highlights the key property of complex conjugates: their product eliminates the imaginary component.