Perform the indicated operation(s) and write the result in standard form. (2 - 3i)(1 - i) - (3 - i)(3 + i)

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First, apply the distributive property (FOIL method) to multiply the complex numbers in each product separately. For the first product, expand $(2 - 3i)(1 - i)$ by multiplying each term in the first parenthesis by each term in the second parenthesis.

Similarly, expand the second product $(3 - i)(3 + i)$ using the distributive property or recognize it as a difference of squares since it is of the form $(a - b)(a + b)$.

After expanding both products, simplify each expression by combining like terms, remembering that $i^2 = -1$.

Subtract the second simplified expression from the first simplified expression as indicated by the problem.

Finally, combine the real parts and the imaginary parts separately to write the result in standard form $a + bi$, where $a$ and $b$ are real numbers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Standard Form of a Complex Number

The standard form of a complex number is written as a + bi, where a is the real part and b is the imaginary coefficient. After performing operations, the result should be simplified and expressed clearly in this form.

Subtraction of Complex Numbers

Subtracting complex numbers involves subtracting their real parts and their imaginary parts separately. This operation is straightforward once both complex numbers are expressed in standard form.

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