We're asked to multiple the complex number 1 minus 3 i times the complex number 2 plus 5 i. And the general idea here is you can multiply these complex numbers like you would have multiplied any traditional binomial you just have to remember this isn't a variable this is the imaginary unit i, or it's just i but we can do that in two ways we can just do distributive properties twice which I like a little bit more just because it actually, you're doing it from a fundamental principle, it's nothing new or you could use FOIL, which you also used when you first multiplied binomials, and I'll do it both ways. So you could view this as just a number, 1 minus 3 i and so we can distribute it over the two numbers inside of this expression. So when we're multiplying times the entire expression We can multiply 1 minus 3 i times two, and 1 minus 3 i times 5 i. So let's do that. So this can be rewritten as 1 minus 3 i times 2, I'll write the 2 out front. Plus 1 minus 3 i, times 5 i. All I did is a distributive property here. All I said is, look if I have a times b plus c, this is the same things as: ab plus ac. I just distributed the a on the b or the c. I just distributed the 1 - minus 3 i on the 2 and the 5 i. And then I can do it again. I have a 2 now times 1 minus 3 i. I can distributed it. 2 times 1 is 2. 2 times negative 3 i is negative 6 i And over here I'll do it again. 5 i times 1, so it's plus, 5 i times one is 5 i And then 5 i times negative 3 i, so let's be careful here 5 times negative 3 is negative 15, and then I have an i times an i Right, I'm multiplying. Let me do this over here. Five i times negative 3 i This is the same thing as 5 times negative 3 times i times i. So the 5 times the negative 3 is negative 15, and then we have i times i, which is i squared. Now we know what i squared is. By definition, i squared is negative 1. i squared by definition is negative 1. So negative 15 times negative 1, well that's the same thing as positive 15. So this can be rewritten as 2 minus 6 i plus 5 i Negative 15 times negative 1 is positive 15 Now we can add the real parts We have a 2 and we have a positive 15 So 2 plus 15, and we can add the imaginary parts We have a negative 6, or negative 6 i I should say and then we have plus 5 i. So plus 5 i And 2 plus 15 is 17 And if I have negative 6 of something plus 5 of that something What do I have? Or if I have 5 of that something, and I take 6 of that something away, then I have negative 1 of that something. Negative 6 i plus 5 i is negative 1 i Or I could just say minus i. So in this way I just multiplied these two expressions or these complex numbers, really I multiplied them just using the distributive property twice. You could also do it using FOIL. And I'll do that right now really fast. It is a little bit faster, but it's a little bit mechanical So you might forget why you're doing it in the first place But at the end of the day, you are doing the same thing here You're essentially multiplying every term of this first number or every part of this first number, times every part of the second number. And FOIL just makes sure that we're doing it. Let me just write FOIL out here, which I'm not a huge fan of, but I'll do it just in case that's the way you're learning it. So FOIL says let's do the first numbers, let's multiply the first numbers so that's going to be the 1 times the 2. So 1 times the 2. That is the "F" in FOIL. Then it says, let's multiply the outer numbers times each other, so that's 1 times 5 i. So plus 1 times 5 i. This is the "O" in FOIL. The outer numbers. Then we do the inners numbers. Negative 3 i times 2. So this is negative 3 i times 2. This is, those are the inner numbers. And then we do the last numbers. Negative 3 i times 5 i. So negative 3 i times 5 i. These are the last numbers. That's all that FOIL is telling us. It's just making sure we're multiplying every part of this number, times every part of that number. And then when we simplify it: 1 times 2 is 2 1 times 5 i is 5 i. Negative 3 i times 2 is negative 6 i And negative 3 i times 5 i, well we already figured out what that was, negative 3 i times 5 i turns out to be 15. Negative 3 times 5 is negative 15. But i times i is negative 1 negative 15 times negative 1 is positive 15. Add to the real parts: 2 plus 15, you get 17 Add to the imaginary parts: you have 5 i minus 6 i. You get negative i And once again, you get the exact same answer.