In Exercises 21–28, divide and express the result in standard ...

Question

In Exercises 21–28, divide and express the result in standard form. (2 + 3i)/(2 + i)

Accepted Answer

Welcome back. I am so glad you're here. We are given these complex numbers and told to divide them. So we are given four plus 11. I divided by four minus I. Alright well when we have complex numbers when we have this I term here and we see it in the denominator of a fraction. We want to get it out of the denominator of the fraction. No imaginary numbers in the denominator. So in order to do that we're going to multiply the denominator by its complex conjugate. Well, we recall from previous lessons on the complex conjugate that when we have a complex number in its standard form. So if it is in the standard form of A plus B. I. A. Is the regular number that we're used to and the B. I. Is our complex number, our imaginary number. We're going to multiply that by its complex conjugate. Which is keeping the regular number, keeping the A. And then swapping the sign of our complex imaginary number. So it's going to be minus B. I. So looking at four minus I. What is that complex conjugate? Well we keep that four term that's R. A. So stays four and then we're going to switch the sign of our imaginary term. So it's now going to be four plus I. And it's important to note that what we do to the denominator we must do to the numerator as well because when we multiply something by four plus I divided by four plus I we're just multiplying it by one. It's like we're barely doing anything to it at all. But we're getting that I out of the denominator. So now let's do that. Let's get the eye out of the denominator. So we're going to foil this. We are going to foil both the numerator and the denominator. I'll start with the numerator. So doing our first four times four gives us 16 and our Outsides four times I gives us plus four I. And our insides 11 I times four is plus 44 I And our lasts 11 I times I is going to be plus I squared. There's our numerator all foiled out. And now let's foil out the denominator four times four is again still 16. And our Outsides four times I is plus four. I our insides negative I times four gives us a minus for I. And our last negative I times I is going to be minus I squared. Now we can combine our like terms in the middle of both the numerator and the denominator. So the numerator will read 16 and then plus four. I plus 44 eyes plus 48. I plus 11. I squared divided by 16. Well plus four I minus four. I those eyes cancel out. Hey, there's that beauty of the complex. Conjugate. We got rid of two of those eyes in the denominator and then minus I squared. But hang on a second. I thought that I said that we'll get rid of the eyes in the denominator. Well we can we recall from previous lessons that by definition I squared equals negative one. So wherever we've got an I. Squared we can replace that with a negative one. So let's do that. 16 plus 48. I plus 11 times not I squared times negative one. All over 16 minus not I squared it's negative one. And now we can do that work multiplying by negative 1. 11 times negative one. That's minus 11 and minus negative one. That's going to be a plus one in the denominator. So now this will be equal to combining our like terms we have 16 minus 11. 16 minus 11 is five. So we have five plus 48 I in the numerator and the denominator. We have 16 combining these like terms 16 plus 1. 16 plus one is 17. So we have five plus 48. I divided by 17. While we look at our answer choices. And this matches with answer choice B. Well done we'll catch you on the next one.

In Exercises 21–28, divide and express the result in standard form. (2 + 3i)/(2 + i)

Key Concepts

Complex Numbers

Division of Complex Numbers

Standard Form of Complex Numbers

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