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

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Recall that to multiply two complex numbers in the form \((a + bi)(c + di)\), you use the distributive property (FOIL method): multiply each term in the first complex number by each term in the second complex number.

Apply the distributive property to \((7 - 5i)(-2 - 3i)\): multiply \(7\) by \(-2\), \(7\) by \(-3i\), \(-5i\) by \(-2\), and \(-5i\) by \(-3i\).

Write out the products explicitly: \(7 \times (-2)\), \(7 \times (-3i)\), \(-5i \times (-2)\), and \(-5i \times (-3i)\).

Simplify each product, remembering that \(i^2 = -1\). For example, \(-5i \times -3i = 15i^2\), which can be rewritten using \(i^2 = -1\).

Combine the real parts and the imaginary parts separately to write the product in standard form \(a + bi\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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