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

Question

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

(2 + 3i)/(2 + i)

Accepted Answer

Welcome back. I'm so glad you're here. We've got a couple of complex numbers and we're going to divide them. So let me rewrite those. We've got four plus 11 I and the numerator. And for -1. In the denominator. How do we go about dividing complex numbers while we've got this I this imaginary number in the denominator. And we don't want that. We don't want imaginary numbers in the denominator. So what we're going to do is we're going to multiply both the numerator and the denominator by what's called the complex conjugate. We recall from previous lessons that the complex conjugate keeps this first term the same. So that's going to stay at four, keeps this last term the same. That's going to stay in I and it switches the sign in the middle. So now it's four plus I. And what we do to the denominator we have to do to the numerator because in effect a number divided by itself. Well that's just one. So we're only multiplying it by one. We're not doing anything too fancy here. And now we can just go ahead and start foiling. So in the numerator we've got four plus I times four plus I in our firsts give us 16. 4 times four is 16. Our Outsides four times I give us plus four I our insides 11 I times four give us plus 44. I and our last 11. I times I gives us plus 11 I squared Great one down. Now let's work on the denominator. We've got four minus I times four plus I. And this is where we get to see how cool the complex conjugate is. So our first four times four is again 16. Our outsides four times I is plus four. I again now for our insights negative I times four is minus for I. In our last negative I times positive I is minus I squared. Alright we can combine like terms in the middle here for I plus 44 I and plus four I minus four. I. So now we've got in the numerator 16 plus four I plus I is 48 I plus 11 I squared. And in the denominator we've got 16 and complex conjugate. It's for I plus negative for I. Those cancel out. So we don't have those middle terms and now it's 16 minus I squared. And here again is where complex conjugate are so cool because we know by definition from previous lessons we recall that I squared equals negative one. So we can replace both of these I squared with negative one. In the numerator we've got 16 plus 48 I plus 11 times negative one. And in the denominator we've got 16 minus negative one. Now we can go ahead and do some simplifying. We've got in the numerator 16 plus 48 I minus 11. And in the denominator 16 minus and negative one is plus one and doing some simplification here combining like terms we've got 16 minus 11. Well that's five plus 48 I and in the denominator 16 plus one, that's 17. We look at our answer choices and this matches answer choice B. Well done, we'll catch you on the next one.

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