(1−i)(1+2i)−(1+4 Zi)(1+2i)

3 min read Jun 16, 2024

Simplifying Complex Expressions: (1−i)(1+2i)−(1+4 zi)(1+2i)

This article will guide you through simplifying the complex expression: (1−i)(1+2i)−(1+4 zi)(1+2i).

Understanding the Steps

The simplification process involves applying the distributive property and then combining like terms. Remember that the imaginary unit, "i", is defined as the square root of -1 (i² = -1).

Step 1: Expanding the Products

First, we need to expand both products in the expression using the distributive property (also known as FOIL):

(1−i)(1+2i): (1 * 1) + (1 * 2i) + (-i * 1) + (-i * 2i) = 1 + 2i - i - 2i²
(1+4 zi)(1+2i): (1 * 1) + (1 * 2i) + (4zi * 1) + (4zi * 2i) = 1 + 2i + 4zi + 8zi²

Step 2: Simplifying using i² = -1

Now we replace all instances of i² with -1:

(1−i)(1+2i): 1 + 2i - i - 2(-1) = 1 + 2i - i + 2
(1+4 zi)(1+2i): 1 + 2i + 4zi + 8(-1) = 1 + 2i + 4zi - 8

Step 3: Combining Like Terms

Finally, we combine the real and imaginary terms separately:

(1−i)(1+2i): (1 + 2) + (2i - i) = 3 + i
(1+4 zi)(1+2i): (1 - 8) + (2i + 4zi) = -7 + (2 + 4z)i

Step 4: Combining the Results

Now, we can combine the simplified results from the two products:

(1−i)(1+2i)−(1+4 zi)(1+2i) = (3 + i) - (-7 + (2 + 4z)i)

Subtracting the second expression from the first, we get:

(3 + i) - (-7 + (2 + 4z)i) = 3 + 7 + i - (2 + 4z)i = 10 - (1 + 4z)i

Conclusion

Therefore, the simplified form of the complex expression (1−i)(1+2i)−(1+4 zi)(1+2i) is 10 - (1 + 4z)i.