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### Course: Algebra (all content)>Unit 10

Lesson 29: Polynomial identities with complex numbers

# Complex numbers & sum of squares factorization

Learn how expressions of the form x^2+y^2 can be factored into linear factors. This would not be possible without the aid of complex numbers! Created by Sal Khan.

## Want to join the conversation?

• how is this whole thing useful . it just makes things more complicated
• There are many reasons we might need to fully factor a polynomial, and this provides a way of giving a more complete answer than "no real solutions". Even a simple quadratic can be very important to describe or model a phenomenon, yet not have real solutions. This method provides a way of stating all the roots. It is definitely important to be able to describe the behavior of polynomials in many careers.

As an example, think of a quadratic equation such as y = x²- 2x - 6
It has two real roots, and it can describe someone's business or scientific phenomenon.

Compare that to y = x²- 2x + 10
This one looks just as promising, but it has no real roots. You might think of it as a loser, therefore. But, if it described a business, it would be one that would never experience a loss! The value of the function never goes below zero. It is useful to be able to figure out why one business model works and one doesn't, or why one model fits a scientific phenomenon, while the other would cause a space vehicle to spin out of control.
• Couldn't you factorize (x^2 + y^2) as (x + y)^2 - 2xy or (x - y)^2 + 2xy?
• Yes and no. Those are correct (and sometimes useful) expressions, but they aren't "factored" because they are not written as a product of factors -- each involves the sum of two terms.

You could factor it as (x + yi)(x - yi), using complex numbers.
• When you write "i", does it need to be written in a different/special way ?
• A lower case i is sufficient, though many people make this lower case i in italic or cursive font - notice the little curl, or serif, that Sal puts on the bottom part of the i he writes.
• Is there a video series or a module on imaginary and complex numbers on Khan Academy as I would like to learn more about them? By the way I am watching this on the Algebra 2 playlist.
• Can you tell why do we use 'i' in these cases? Is it just for a reference or can we use other letters too?
• So it basically equals z times its conjugate, where z = x + yi.
• Why we can't just factor it like this
x^2+y^2=x(x+y^2/x) ?
In this case it's a product of factors, aren't it?
• Yes, you can do that, but one of your resulting factors not only still have a variable squared, but also has a variable to the `-1` power, so it can be seen as even more complex than the original expression.

Factorization is commonly used to simplify a expression, the video shows a way in which you end up with two linear factors, which are simpler than a quadratic expression.
• I've taken the sum of cubes rule which is (a+b)(a^2-ab+b^2) ... Is there a similar rule for the sum(or difference) of ''4th power'' terms?
• The difference of fourth powers can be treated as the difference of squares. Since a^4 = (a^2)^2, we can write: a^4 – b^4 = (a^2)^2 – (b^2)^2 = (a^2 + b^2)(a^2 – b^2) = (a^2 + b^2)(a + b)(a – b). Now, a^2 + b^2, technically, can be factored over the irrational numbers: a^2 + b^2 = a^2 + 2ab + b^2 – 2ab = (a + b)^2 – 2ab = (a + b – sqrt(2ab))(a + b + sqrt(2ab)), or we can factor it over the complex numbers as shown in Sal’s video.
The sum of fourth powers can be treated as the sum of squares and factored over the irrational numbers too: a^4 + b^4 = a^4 + 2*a^2*b^2 + b^4 – 2*a^2*b^2 = (a^2 + b^2)^2 – 2(ab)^2 = (a^2 + b^2 – sqrt(2)ab)(a^2 + b^2 + sqrt(2)ab). Or we can factor it over the complex numbers.
• Is 2(x+y)^2 considered a sum of squares?
(1 vote)
• Sum or squares would be: x^2 + y^2
Your factors will create a perfect square trinomial with an additional common factor of 2
Hope this helps.