(3+i)(i^2+8i-2)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (3 + i)(i^2 + 8i - 2)

This article will guide you through multiplying the complex numbers (3 + i) and (i^2 + 8i - 2). We'll break down the process and demonstrate the key steps involved.

Understanding Complex Numbers

Complex numbers are expressed in the form a + bi, where:

• a represents the real part.
• b represents the imaginary part.
• i is the imaginary unit, where i^2 = -1.

The Multiplication Process

1. Expand the expression: Begin by using the distributive property (or FOIL method) to expand the product:

   (3 + i)(i^2 + 8i - 2) = 3(i^2 + 8i - 2) + i(i^2 + 8i - 2)

2. Simplify using i^2 = -1: Substitute -1 for i^2 in the expression:

   = 3(-1 + 8i - 2) + i(-1 + 8i - 2)

3. Further Simplification: Distribute and combine like terms:

   = -3 + 24i - 6 - i - 8i^2 - 2i = -3 + 24i - 6 - i + 8 - 2i

4. Final Result: Combine the real and imaginary terms:

   = -1 + 21i

Conclusion

Therefore, the product of (3 + i) and (i^2 + 8i - 2) is -1 + 21i.