Perform the indicated operation(s) and write the result in standard form. (8 + 9i)(2 - i) - (1 - i)(1 + i)

Verified step by step guidance

1. First, apply the distributive property (FOIL method) to multiply the complex numbers in the first product: \((8 + 9i)(2 - i)\). Multiply each term in the first binomial by each term in the second binomial.
2. Next, multiply the complex numbers in the second product: \((1 - i)(1 + i)\). Recognize this as a difference of squares pattern, since \((a - b)(a + b) = a^2 - b^2\).
3. After expanding both products, combine like terms in each product separately. Remember that \(i^2 = -1\), so replace any \(i^2\) terms accordingly.
4. Subtract the result of the second product from the result of the first product by combining the real parts and the imaginary parts separately.
5. Finally, write the resulting expression in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part.

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 their imaginary parts separately. This operation is straightforward once both complex numbers are expressed in standard 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 coefficient of the imaginary part i. Writing results in this form makes it easier to interpret and perform further operations.