Multiply Complex Numbers - Algebra-2

1. Fundamental Concepts

Multiplying complex numbers is similar to multiplying polynomials, following the distributive property (FOIL method). This involves multiplying each term of one complex number with each term of the other, then combining like terms.
Since complex numbers are in the form (where a is the real part, bi is the imaginary part, and i is the imaginary unit with ), it is crucial to convert to during calculations to simplify the result.

Multiplication Rule: For two complex numbers and , their product is: Simplifying using , this becomes: In other words, the real part of the result is the product of the real parts minus the product of the coefficients of the imaginary parts, and the coefficient of the imaginary part is the sum of the cross-products of the terms.
Closure Property: The product of two complex numbers is also a complex number.
Special Cases: The product of pure imaginary numbers (e.g., ) is a real number (), because the product of their coefficients multiplied by eliminates the imaginary unit.

Easy difficulty: Calculate
Solution: Expand using the FOIL method
Substitute :.
Medium difficulty: Calculate
Solution: Expand and combine like terms (since ).
Hard difficulty: Calculate
Solution: Multiply step by step, starting with the first two complex numbers
Step 1: ;
Step 2: .

Step 1: Expand using polynomial multiplication: Apply the distributive property (or FOIL method: First, Outer, Inner, Last) to multiply each term of the two complex numbers crosswise.
Step 2: Combine like terms: Group and combine the real terms (without i) and the imaginary terms (with i) separately, paying attention to calculating the coefficients of the imaginary terms.
Step 3: Simplify : Replace all instances of with , then further combine the real and imaginary parts.
Step 4: Arrange the result: Express the final result in the standard form (real part first, imaginary part second).