In Exercises 1–3, perform the indicated operations and write the result in standard form. (6 − 7i)(2 + 5i)

Verified step by step guidance

1. Identify the expression to be expanded: \((6 - 7i)(2 + 5i)\).
2. Apply the distributive property (FOIL method) to expand the expression: \((6)(2) + (6)(5i) + (-7i)(2) + (-7i)(5i)\).
3. Calculate each term separately: \(6 \times 2\), \(6 \times 5i\), \(-7i \times 2\), and \(-7i \times 5i\).
4. Combine the real parts and the imaginary parts separately: \(12 - 35i^2\) and \(30i - 14i\).
5. Remember that \(i^2 = -1\), so replace \(-35i^2\) with \(35\) and combine all terms to write the result in standard form \(a + bi\).

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, b is the imaginary part, and i is the imaginary unit 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 combine the real and imaginary parts. For example, when multiplying (a + bi)(c + di), you calculate ac, ad, bc, and bd, and then combine like terms, remembering that i^2 = -1 to 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 performing operations on complex numbers, the final result should be simplified to this form, ensuring that the real and imaginary components are clearly separated for clarity and further calculations.