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. Recall that to multiply two complex numbers, we use the distributive property (FOIL method): \((a + bi)(c + di) = ac + adi + bci + bdi^2\).
2. Apply the distributive property to \((6 - 7i)(2 + 5i)\): multiply each term in the first parenthesis by each term in the second parenthesis.
3. Calculate each product: \(6 \times 2\), \(6 \times 5i\), \(-7i \times 2\), and \(-7i \times 5i\).
4. Remember that \(i^2 = -1\), so replace \(i^2\) with \(-1\) in the expression.
5. Combine the real parts and the imaginary parts separately to write the result in standard form \(a + bi\).

Key Concepts

Complex Number Multiplication

Multiplying complex numbers involves using the distributive property (FOIL method) to expand the product. Each term in the first complex number is multiplied by each term in the second, combining like terms and applying the rule i² = -1 to simplify.

Standard Form of a Complex Number

The standard form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary part. After multiplication, the result should be simplified and rearranged to clearly separate the real and imaginary components.

Imaginary Unit Properties

The imaginary unit i is defined such that i² = -1. This property is essential when simplifying products involving i, as it allows conversion of i² terms into real numbers, enabling the expression to be written in standard form.