Problem 11 - Algebra and Trigonometry

Question

Find each product and write the result in standard form. $$ (-5+4 i)(3+i) $$

Step-by-Step Solution

Answer

-19 + 7i

Step 1: Distribute

Multiply each term in the first complex number by each term in the second complex number: \((-5) * 3\), \((-5) * i\), \(4i * 3\), \(4i * i\).

Step 2: Compute Products

Calculate the products found in Step 1: \((-5 * 3) = -15, (-5i) = -5i, 4i*3 = 12i, 4i * i = 4i^2\).

Step 3: Rewrite \(i^2\) as -1

Use the property \(i^2 = -1\) and substitute it: \(4i^2\) becomes \(4*(-1)\) which is \(-4\).

Step 4: Combine Like Terms

Add together the real and the imaginary parts respectively: \(-15 - 4 = -19; -5i + 12i = 7i\).

Step 5: Write in Standard Form

Write the final answer in standard form, combining the values calculated in the previous step: \(-19 + 7i\).