Factor 𝑥 squared plus 𝑦 squared over the complex numbers.
In this question, we’re given an algebraic expression and we’re told to factor this algebraic expression. And we’re told to do this over all of the complex numbers. To do this, let’s start by discussing exactly what this means.
When we’re asked to factor an algebraic expression, we want to write this as the product of more simple algebraic expressions. And in this question, since we’re told we’re allowed to do this over the complex numbers, this means our factors are allowed to include complex numbers.
So let’s start with the algebraic expression we’re asked to factor. That’s 𝑥 squared plus 𝑦 squared. And at first, it doesn’t seem like there’s any easy way to factor this expression. However, there is one thing we can notice about this expression. It is very similar to an expression we do know how to factor. We know how to factor a difference between two squares. We know 𝑥 squared minus 𝑦 squared will be equal to 𝑥 plus 𝑦 multiplied by 𝑥 minus 𝑦.
So we know how to factor a difference between two squares. However, the algebraic expression we’re given is the sum of two squares. So if we know how to factor this expression when we’re taking a difference, let’s try rewriting this expression with a difference. So one thing we can try is to rewrite our algebraic expression as 𝑥 squared minus negative 𝑦 squared.
Now we’re taking the difference between two terms. So we could try using difference between squares on this expression. We can see our first term of 𝑥 squared is a square. However, our second term of negative 𝑦 squared is not a square. So in its current form, we can’t yet apply difference between squares because our second term is not a square.
Therefore, to apply difference between squares to this expression, we’re going to need to find something which squares to give us negative 𝑦 squared. And there’s a few different ways we could find this. One way of doing this is to notice that negative one times 𝑦 squared will be equal to 𝑖 squared times 𝑦 squared. And this is because 𝑖 is the square root of negative one, so 𝑖 squared will be equal to negative one.
Therefore, we can use this to rewrite our algebraic expression as 𝑥 squared minus 𝑖 squared 𝑦 squared. And now we can notice something interesting. In our second term, both of our factors have the same exponent. So either by using our laws of exponents or by writing the product out in full, so that’s 𝑖 squared times 𝑦 squared is equal to 𝑖 times 𝑖 multiplied by 𝑦 times 𝑦, we can rewrite this term as 𝑦 times 𝑖 all squared.
Therefore, by using this, we were able to rewrite the algebraic expression given to us as 𝑥 squared minus 𝑦𝑖 all squared. And now we can see we’ve successfully written this expression as the difference between two squares. So by using difference of squares, we can factor this to give us 𝑥 plus 𝑦𝑖 all multiplied by 𝑥 minus 𝑦𝑖. And this is our final answer.
Therefore, by using a difference between two squares and by factoring over the complex numbers, we were able to show that 𝑥 squared plus 𝑦 squared is equal to 𝑥 plus 𝑦𝑖 all multiplied by 𝑥 minus 𝑦𝑖.