Problem 12 - Algebra and Trigonometry

Question

Find each product and write the result in standard form. $$ (-4-8 i)(3+i) $$

Step-by-Step Solution

Answer

The product is \(-20 - 28i\) when written in standard form

Step 1: Distribute the Real and Imaginary Parts

Distribute the real part of the first complex number to both parts of the second complex number and do the same to the imaginary part. Replace \(i^2\) with -1 because it represents the square root of -1. So we get: \((-4)(3) + (-4)(i) + (-8i)(3) + (-8i)(i)\)

Step 2: Simplify each term

Each term simplifies as follows: \(-12 - 4i - 24i - 8i^2\). Replace \(i^2\) with -1, giving us: \(-12 - 4i - 24i + 8(-1)\)

Step 3: Combine like terms

Combine all of the real parts together and all of the imaginary parts together: \((-12 - 8) + (-4i - 24i) = -20 - 28i\) which is the final product.