From AOPS. If , find . If if A + B = 6 and AB = 3.

Factoring is Awesome9/6/11 (from random tests that I don’t care to disclose at 3:00 am but I still want to credit them) • From AOPS. If , find. • If if A + B = 6 and AB = 3. • Given that 98762 = 97535376, find 98772. • Factor completely: x2 + 2mn – m2 – n2. • If x + y = 4 and xy = 2, then find x6 + y6.

Factoring is Awesome • Evaluate the sum • Simplify completely: . • Find if . • Factor x12 – y12 as completely as possible with integral coefficients and integral exponents. Good luck.

Key First of all, you have an equation. You need to do something with that equation, but what? You are trying to find terms that are squared, so I think your best bet is to square the first equation. What do you get? • From AOPS. If , find. You are used to foiling, right? Here’s a tip for math competitions: When you square binomials like this, it’s usually easier to write it in the form of a2 + b2 + 2ab instead of a2 +2ab +b2. 1 This is what you want!!!

Key Hmmm… it seems like you don’t have any useful information at first, but WAIT. What can you do to that first expression?? Possibly make it one fraction? Let’s try that… • If if A + B = 6 and AB = 3. Yeah. It was that easy. Be on the lookout for this problem. It’s considered a classic  so it may come up from time to time… (especially ciphering and stuff)

Key This is where a little bit of practice comes in handy… For problems like these, you kind of have to guess what method you have to use (if you could consider this a “method”). So let’s try something…. • Given that 98762 = 97535376, find 98772. So if we find x, we will be able add it to 97535376 to get 98772. But how do we find x?? HINT: Do you see a difference of squares anywhere? Pretty number!! Let’s add!!!

Key First, let’s rearrange the terms… What do you notice? • Factor completely: x2 + 2mn – m2 – n2. You have an expanded (m – n)2 ! Let’s factor that out… Notice something else??? AN AWESOME DIFFERENCE OF SQUARES! Now let’s simplify this a bit and make it look nice 

Key Okay. So we have to get to an x6 term… How do we get there?? We could square then cube, or cube then square… Let’s try the second way! • If x + y = 4 and xy = 2, then find x6 + y6. Here’s a useful way to look at the cube of (x + y): Now we square…

Key You have a whole bunch of square roots in the denominator… What are your options? You could find the least common denominator, but that would be unbelievably messy… You could also multiply by the conjugates. Remember those? Here’s an example: • Evaluate the sum Notice the difference of squares? If you multiplied each of the fractions by its conjugate, here’s what you’d get: You can cancel out a lot of stuff  That’s the best part!

Key At first, this seems like a very hard problem, but just think about what we have been doing… We can simplify the first part. Great. But what about the last half? How do 56930 and 56926 relate to 56928? They’re both 2 away… Let’s show that! • Simplify completely: . Yeah. Difference of squares comes up. A lot. Sometime sure cancels  ! Let’s see what we have left… Note: If the square root is there at the beginning of the problem, it’s okay to take just the positive square root. If you had the square root (to solve and equation of something), you need to keep both plus and minus (unless the solution is extraneous ).

Key Okay. So we have to get to an x6 term… How do we get there?? How about we square and then square again  • Find if . Watch the negative! 1 Let’s square again! Isn’t this getting much easier? 1

Key Let’s take this problem step by step… A difference of squares is the easiest to do… • Factor x12 – y12 as completely as possible with integral coefficients and integral exponents. Good luck. Difference of squares Sum of cubes Difference of cubes Sum of cubes Now let’s put all of these factors together…

From AOPS. If , find . If if A + B = 6 and AB = 3.