In Exercises 53–58, perform the indicated operation(s) and write the result in standard form.(2 − 3i)(1 − i) − (3 − i)(3 + i)

Verified step by step guidance

1. Distribute the terms in the first expression: \((2 - 3i)(1 - i)\).
2. Calculate \(2 \cdot 1\), \(2 \cdot (-i)\), \(-3i \cdot 1\), and \(-3i \cdot (-i)\) and combine the results.
3. Distribute the terms in the second expression: \((3 - i)(3 + i)\).
4. Calculate \(3 \cdot 3\), \(3 \cdot i\), \(-i \cdot 3\), and \(-i \cdot i\) and combine the results.
5. Subtract the result of the second expression from the result of the first expression and simplify to write in standard form \(a + bi\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding how to manipulate complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Multiplication of Complex Numbers

To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials) and combine like terms. For example, when multiplying (a + bi)(c + di), you calculate ac, adi, bci, and bdi^2, remembering that i^2 = -1. This process is crucial for simplifying expressions involving complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. When performing operations with complex numbers, the final result should be expressed in this form for clarity. This involves combining real parts and imaginary parts separately and ensuring that any instances of i^2 are replaced with -1 to maintain the standard format.