What is negative nine plus five 𝑖 multiplied by three minus 𝑖?
In this question, we’re asked to evaluate the product of two complex numbers, and we can see these are both given in algebraic form. Remember, the algebraic form of a complex number is the form 𝑎 plus 𝑏𝑖 where 𝑎 and 𝑏 are real numbers. So to multiply these two complex numbers together, we need to notice something interesting about the form they’re given in. Inside each pair of parentheses, we have two terms. These are two binomial expressions, so we can multiply two binomials together by using the FOIL method.
So let’s start by multiplying the first two terms of our parentheses together. And that gives us negative nine multiplied by three. Next, the FOIL method tells us we need to add on the product to the outer two terms in our parentheses. This means we need to add on negative nine multiplied by negative 𝑖. The next step in the FOIL Method is to add on the products of our two inner terms. And that’s five 𝑖 multiplied by three. Finally, the last step is to add on the product of the two last terms in our parentheses. And that’s five 𝑖 multiplied by negative 𝑖.
Now that we’ve distributed our parentheses, we can simplify each of these products separately. First, negative nine multiplied by negative three is equal to negative 27. Next, we have negative nine multiplied by negative 𝑖. Well, a negative times a negative is a positive, so this is going to be equal to nine 𝑖. Our third term is five 𝑖 multiplied by three. We multiply the five and the three together to get 15, giving us that this is going to be equal the 15𝑖. The fourth term is slightly more difficult. We have five 𝑖 multiplied by negative 𝑖. First, we need to multiply the negative one and the five together to give us negative five. However, we still have 𝑖 multiplied by 𝑖, and we’ll write this as 𝑖 squared.
So far, we’ve shown that the product of these two complex numbers is equal to negative 27 plus nine 𝑖 plus 15𝑖 minus five 𝑖 squared. However, we can simplify this further. We notice both our second and third term could be added together. To add these two terms together, we just need to add the coefficient of 𝑖 together. And nine plus 15 is equal to 24, so these add together to give us 24𝑖.
But we can simplify this even further. We need to notice something about our third term. We can see our third term contains a factor of 𝑖 squared. And we know that 𝑖 is the square root of negative one. So 𝑖 squared is equal to negative one. So we can replace 𝑖 squared in this expression with negative one, giving us negative 27 plus 24𝑖 minus five times negative one. And of course, subtracting five times negative one is the same as adding five. And then we have negative 27 plus five, which is equal to negative 22, giving us negative 22 plus 24𝑖, which is our final answer. Therefore, we were able to show that negative nine plus five 𝑖 multiplied by three minus 𝑖 is equal to negative 22 plus 24𝑖.