Factoring Special Case Polynomials 05:37 minutes
Transcript Factoring Special Case Polynomials
Daniella Feinberg, who recently retired from the fur business, and her husband are trying to enjoy a relaxing afternoon in front of their pool. She wants her husband, who makes a handsome living designing pools, to design one so she can throw a pool party...
What's this? The neighbor's dog just came through the fence and it looks like he has plans to do a little bit of surfing...Uhoh... it looks like this pug's doggie instincts are taking over and he's ripping up the inflatable pool! I guess he's all about that pug life. Daniella's up in arms about the neighbor's dog. I guess her husband has his work cut out for him
Side lengths of the pool
Mr. Feinberg suggests that they build a square pool with side lengths 'a'. Using what we learned about area, we know we can find the area by multiplying the two side lengths together. Doing so gives us a², but Daniella doesn't like this option because it's too "normal". To make the pool less "normal", Mr. Feinberg suggests side measurements of (a  b), giving them a pool area of (ab)(a  b) or (a  b)².
The FOIL method
Being the consummate professional, Mr. Feinberg uses the FOIL method to figure out exactly what he's dealing with. First he subtracts the b term from his existing plans. Since he already knows a(a) is a², he can concentrate on the other terms. a(b) is ab and since we have this twice, we can write it here...and here.
Finally, he has two 'b's that he needs to multiply together, giving him +b². Now he just adds all the terms together and combines like terms, leaving him with a²  2ab + b². Daniella thinks it'd be silly if she threw a pool party with this small of a pool, so then Mr. Feinberg suggests (a + b)(a + b) to give Daniella the biggest pool that'll fit in their yard.
Write this mathematically
We can write this mathematically as (a + b)(a + b) or (a + b)². Mr. Feinberg's training is coming in handy, so when he applies the FOIL method to (a + b)², he knows he'll get a positive a² and b² and 'ab' twice gives him 2a making his final expression a² + 2ab + b². Since Daniella doesn't want to give up her rose garden, and since square pools are SO five minutes ago, she rejects this option as well.
Then, in a stroke of genius, Mr. Feinberg comes up with (a + b)(a  b). Mr. Feinberg uses the FOIL method one last time and gets a² one '+ab' one 'ab' and a b². The two 'ab' terms cancel out, leaving Mr. Feinberg with a²  b². He draws it out for Daniella who looks at him quite curiously. She shows him that he's only cut out a little block from the area, not to mention the pool looks funny!
The area
Eureka again! Mr. Feinberg can move this piece here to make the pool not look as funny! All this talk of As and Bs is making Daniella a bit dizzy. She wants to know the area of the pool so that she can start planning the party. The Special Case Polynomials that Mr. Feinberg suggested were: a², (a  b)², (a + b)², and (a + b)(a  b).
Use PEMDAS for calculation
We can use any numbers for 'a' and 'b', but we'll use 10 feet for 'a' and 5 feet for 'b'. When we plug numbers in, instead of applying FOIL to our expression, we can use PEMDAS to make our calculations easier. Plugging 10 feet in for 'a' and 5 feet for 'b' gives us 100 ft.² for a² 25 ft.² for (a  b)² 225 ft.² (a + b)² and 75 ft.² for (a + b)(a  b). Good things usually come in threes, and Mr. Feinberg has had another brilliant idea...
All in a hard day's work! They're finally done! Now Daniella can throw her blowout pool party! What's this?!? Looks like a pugly situation...

Introduction to Polynomials – Naming Polynomials by Number of Terms

Adding Polynomials

Multiplying Polynomials

Multiplying Special Case Polynomials

Factoring out the GCF

Factoring Trinomials with a = 1

Factoring Trinomials with a ≠ 1

Factoring Special Case Polynomials

Factoring by Grouping

Subtracting Polynomials