(5-2i)(2+7i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (5-2i)(2+7i)

This article will guide you through the multiplication of complex numbers, specifically the product of (5-2i) and (2+7i).

Understanding Complex Numbers

Complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).

The Multiplication Process

To multiply complex numbers, we use the distributive property, similar to multiplying binomials:

1. FOIL Method: "First, Outer, Inner, Last"

   • First: (5)(2) = 10
   • Outer: (5)(7i) = 35i
   • Inner: (-2i)(2) = -4i
   • Last: (-2i)(7i) = -14i²

2. Simplify: Remember that i² = -1, so we substitute:

   • 10 + 35i - 4i - 14(-1)

3. Combine Real and Imaginary Terms:

   • (10 + 14) + (35 - 4)i

4. Final Result:

   • 24 + 31i

Therefore, the product of (5-2i) and (2+7i) is 24 + 31i.

Key Points

• Complex Multiplication: The product of two complex numbers is also a complex number.
• FOIL Method: This method ensures that all terms are multiplied.
• Simplifying with i²: Remember to substitute i² with -1.

By following these steps, you can confidently multiply complex numbers and arrive at the correct answer.