In Exercises 9–20, find each product and write the result in standard form. (7 - 5i)(- 2 - 3i)

Verified step by step guidance

1. Identify the expression to be multiplied: \((7 - 5i)(-2 - 3i)\).
2. Apply the distributive property (also known as the FOIL method for binomials): First, multiply the first terms: \(7 \times -2\).
3. Next, multiply the outer terms: \(7 \times -3i\).
4. Then, multiply the inner terms: \(-5i \times -2\).
5. Finally, multiply the last terms: \(-5i \times -3i\).

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 and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding 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) to each part of the numbers. This involves multiplying the real parts and the imaginary parts separately, and then combining like terms, remembering that i^2 equals -1, which helps simplify the result.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. When multiplying complex numbers, the final result should be simplified to this form, ensuring that the real and imaginary parts are clearly separated for clarity and ease of interpretation.