(x-1-i).jpeg "(x-1+i)(x-1-i)")
Multiplying Complex Numbers: (x-1+i)(x-1-i)
This expression involves multiplying two complex numbers. Let's break down the process and understand the result:
Understanding Complex Numbers
Complex numbers are expressed in the form a + bi, where:
- a represents the real part.
- b represents the imaginary part.
- i is the imaginary unit, defined as the square root of -1 (i² = -1).
Multiplying the Expression
We'll use the distributive property (also known as FOIL) to multiply the complex numbers:
(x-1+i)(x-1-i) = (x-1)(x-1) + (x-1)(-i) + (i)(x-1) + (i)(-i)
Now, let's simplify each term:
- (x-1)(x-1) = x² - 2x + 1
- (x-1)(-i) = -ix + i
- (i)(x-1) = ix - i
- (i)(-i) = -i² = -(-1) = 1
Combining the terms:
(x² - 2x + 1) + (-ix + i) + (ix - i) + 1 = x² - 2x + 2
The Result
Therefore, the product of (x-1+i)(x-1-i) is x² - 2x + 2. Notice that the result is a real quadratic polynomial with no imaginary component.
Why is the Result Real?
The reason we get a real result is due to the special nature of the two complex numbers being multiplied. They are complex conjugates.
Complex conjugates have the same real part but opposite imaginary parts. When multiplied, the imaginary terms cancel out, leaving only the real terms.