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. Recall that to multiply two complex numbers, use the distributive property (FOIL method): for \((a + bi)(c + di)\), the product is \((ac - bd) + (ad + bc)i\).
2. First, multiply the complex numbers in the first product: \((2 - 3i)(1 - i)\). Apply the distributive property carefully.
3. Next, multiply the complex numbers in the second product: \((3 - i)(3 + i)\). Remember that \((a - bi)(a + bi) = a^2 + b^2\) because it is a difference of squares.
4. After finding both products, subtract the second product from the first as indicated: \((2 - 3i)(1 - i) - (3 - i)(3 + i)\).
5. Finally, combine like terms (real parts together and imaginary parts together) to write the result in standard form \(a + bi\).

Key Concepts

Complex Number Multiplication

Multiplying complex numbers involves using the distributive property (FOIL) and applying the rule i² = -1. Each term in the first complex number is multiplied by each term in the second, then like terms are combined to simplify the expression.

Complex Number Subtraction

Subtracting complex numbers requires subtracting their real parts and imaginary parts separately. This operation is straightforward once both complex numbers are expressed in the form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Writing the result in this form means simplifying the expression so that it clearly shows the real and imaginary components.