High School: Algebra
Arithmetic with Polynomials and Rational Expressions HSA-APR.C.4
4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
With the increase in technology and this huge new thing called the Internet, identity theft has become a worldwide problem. For this reason, it is paramount to keep important information such as addresses and telephone numbers as private as possible when online. If not, you might have thieves coming to your home and stealing your polynomial identities.
Sure, your identity should be kept safe too, but we were talking more about polynomial identities. A polynomial identity is just a true equation, often generalized so that it can apply to more than one situation. For instance, (x2 + y2)2 = (x2 – y2)2 + (2xy)2 is true for any x and y. But how can we be sure that's true unless we prove it?
Students should be able to prove identities by showing that one side of an equation is equal to the other. That just takes the same skills they use to organize equations and expressions. They can leave their room-cleaning skills at home, since we all know how organized those are.
To prove polynomial identities, students can either work from one side of the equation to try to derive the other side, or can work from both sides to get the same thing. Both are just fine as long as both sides end up being exactly the same. The goal is to make both sides identical.
Students should also understand the use of certain identities. Sometimes, memorizing an identity is quicker than working out the algebra longhand, and other times the identity applies to a particular context.
We can prove the identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 by manipulating the equation step by step, showing our work as we go. We'll change only one side of the equation and see if we can change it into the other side.
(x2 + y2)2
= (x2 + y2)(x2 + y2)
= x4 + x2y2 + y2x2 + y4
= x4 + 2x2y2 + y4
= x4 – 2x2y2 + 4x2y2 + y4
= (x4 – 2x2y2 + y4) + 4x2y2
= (x2 – y2)2 + (2xy)2
There. Since both sides are equal, we can call this equation an identity. The identity is true for all values of x and y. For x = 4 and y = 2? Yep. How about for x = -1 and y = 398,128? You betcha.