In Exercises 53–58, perform the indicated operation(s) and wri...

Question

In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 − 3i)(1 − i) − (3 − i)(3 + i)

Accepted Answer

Hello, everyone. We want to know which of the following expressions is equivalent to open parentheses. Four minus five. I closed parentheses multiplied by the parentheses. Nine minus I closed parentheses minus the parentheses. 11 minus I closed parentheses multiplied by the parentheses. 11 plus I closed parentheses. Answer choices are a negative 102 minus 57 I B 53 plus 36 I C negative minus 49 I D 82 plus 41 I. So to begin with, I'm going to rewrite our expression. So four minus five, I is a complex number in parentheses nine minus I is another complex number that's multiplied by it. And then we subtract the complex number 11 minus I and multiply it by the complex number 11 plus I. So when multiplying complex numbers, we treat this just like if we were multiplying binomials. So we are gonna go back to that idea of foil. So first, outer, in our last, I'm gonna do the multiplication, then the subtraction. So first, in my first set, four multiplied by nine outer would be four, multiplied by negative I, so negative four I in negative five, I multiplied by nine. So negative 45 I and last negative five, I multiplied by negative, I should be positive five I squared. I'm going to combine my like terms here and simplify before I do the next set of multiplication. So far my like terms I have negative four I and negative 45. I, so I have 36 minus 49 I plus and it's five multiplied by I squared. Well, I recall that I squared is the same as negative one. So we're gonna do five multiplied by negative one, which is negative five. So I'm gonna add that to 36 I have 31 minus 49. I as my first complex number, I'm gonna bring my minus sign straight down because I will deal with that after I'm done multiplying and now I'm gonna foil 11 minus I multiplied by 11 plus I, if you look at this and notice that this is a product of a sum and a difference and recall what the shortcut is, please feel free to use it. Recalling the shortcut would be I square the first term. So 11 multiplied by 11. So 11 squared is 121 and I square. The last term. So I squared is I squared and it's gonna be the difference of two squares. So we put a minus sign between them. If you used foil, you would get the same result. But you'd have middle terms that canceled out. So now I have 121 minus I squared. We're calling again, I squared is equivalent to negative one. This is the same as saying 121 plus a negative one. I'm sorry, plus a positive one or minus a negative one. So this will be 122 which I will bring down here. So now our simplified expression and I had put parentheses because I thought I'd need parentheses on my second term. But I don't, our simplified expression is 31 minus I minus 122 combining like terms. I'm going to do 31 minus 122 and that is negative 91 and 49. I doesn't have anything to be combined with. So it's negative 91 minus 49 I, and that is answer choice C have a nice day.

In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 − 3i)(1 − i) − (3 − i)(3 + i)

Key Concepts

Complex Number Multiplication

Complex Number Subtraction

Standard Form of a Complex Number

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