Find each quotient. Write answers in standard form. (1-3i) / (...

Question

Find each quotient. Write answers in standard form. (1-3i) / (1+i)

Accepted Answer

Hello, everyone. We've been asked to perform division and express the quotient in standard form. We are dividing the parentheses 14 minus four. I by the parentheses, seven plus I, our answer choices are a 51 28th minus 11 20 eights I B 28th plus 11 28th. I C 47 25th minus 21 25th I D 47 25th plus 21 25th. I, the first thing I will do is rewrite this division problem as if it's a fraction. So 14 minus four, I is my numerator seven plus I is my denominator. I recall to get the imaginary number out of the denominator. I need to multiply top and bottom by a conjugate. A conjugate has the same terms as our denominator, but the opposite sign on the imaginary part. So our denominator was seven plus I, our conjugate is seven minus I and we are going to multiply this all the way through for the numerator. So it is very much like if we had foil. So I'm gonna treat it the same way. First is seven multiplied by 14, which is 98 outer negative I multiplied by 14 would be minus 14 I or negative 14, I inner negative four, I multiplied by seven is negative 28 I and the last would be negative four, I multiplied by I which is gonna be plus four I squared. We'll simplify that. In the next step. For my denominators, the conjugates purposely have opposite middle signs. So all we need to do is square. The first term. So seven squared is 49 and square. The last term which is going to be I squared and it would be a minus in between them. Next, we're gonna simplify this and in simplifying this recall that I squared is the same as negative one. So we have minus 42 I plus four times negative one because I'm putting that in for I squared and my denominator would be 49 minus negative one. So I'm gonna simplify it again now that I've put the negative ones in, in the numerator four multiplied by negative one is negative four. So 98 plus negative four is minus 42. I in my denominator, 49 minus a negative one is like 49 plus one, which is 50. So at this step, I have 94 minus 42 I over 50 we want this to be in standard form. So I'm gonna separate this into the real part and the imaginary part. So I'm gonna make it two separate fractions So I'll have 50th minus 42. I over 50 I want to reduce these fractions if I can. And it looks like they should be able to be reduced by two. So 94 divided by two is 47 divided by two is 25. So our first fraction is 47 25th minus 42 divided by two is 21. And the denominator 50 divided by two is 25. And we could put the I as if it's multiplied by the entire fraction. It does not need to just be in the numerator. This is our solution. This is the quotient. We have 47 25th minus 25th I, and this matches answer choice C have a nice day.

Find each quotient. Write answers in standard form. (1-3i) / (1+i)

Key Concepts

Complex Numbers and Standard Form

Division of Complex Numbers

Complex Conjugate

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