Thursday, September 14, 2006
posted by Jeff R. at 7:46 p.m.

First thing's first, we started the class by having Ms. Armstrong to post the answer key for the S1 Review at the back of the room. So if you need to correct your work, check it out sometime. Anyway, we were in groups once again. Ms. Armstrong turned on the projector and then we started taking down notes on "Multiplying Polynomials".

*Here are the notes incase you were lazy in class and didn't copy them :P

Multiplying Polynomials

= 12
Multiplying means "finding the area of a rectangle".

Example:
Find the area.
Now, there are 3 ways that you can do to find the area or to multiply the polynomials. It depends on which one you are most comfortable with:

The first way of multiplying polynomials is to use the "Tic Tac Toe" format. Using the above question, it will look like the image on the left. The yellow "x" is the multiplication sign. The "x+1" represents the width, and the "2x+2" represents the length of the rectangle. So it'll be like, (2x)(x) = 2x2 and so forth. Then you add the like terms which are the ones that are circled.

The second way is using the "Algetiles". This format is like the Tic Tac Toe format except it is presented in images. If you don't know what the images represents, see the previous post.

The last one is everybody's favourite, the "Distributive Property". You first do the red one then the yellow then the green then the blue. So the answer would be:
= 2x2 + 2x + 2x + 2
= 2x2 + 4x + 2 (the like terms were added to simplify)

Dividing Polynomials

Dividing polynomials isn't that hard once you got how to do it properly. On an equation like this:
16x4y5 - 8x8y9
------------------------------
4xy4

Reminder that the denominator applies to all the polynomials on top. So here's how you divide polynomials:

i. Divide the coefficient (the number before the variable) on the denominator from the coefficient on the numerator. (Eg: 16 divided by 4)
ii. Subtract the exponents
iii. Then do the same thing with the second monomial.

**The final answer will look like this: 4x3y - 2x7y5

Like the multiplication of polynomials, dividing polynomials also have different ways to represent the equation:

i. (4x2+x3+5+2) division sign (x+2)
ii. (4x2+x3+5+2) / (x+2)
iii. Divide (4x2+x3+5+2) by (x+2)
iv. Find the length of a rectangle if the AREA is (4x2+x3+5+2) and the width is (x+2).

Long Division:
Before you solve an equation in long division, you must make sure that your terms are in descending order. If it's 4x2+x3+5+2, it should be x3+4x2+5+2, since the exponent "3" is higher than the exponent "2" even though it has a coefficient before it and the other one doesnt.

Long Division in polynomials is the same thing as dividing long division normally.

Between the notes, we've had an activity. The activity was to find the area of this:

Our Solution

My group started to find the area of the "whole rectangle" (on the left side, picture 2). Then subtracted the area of the small rectangle (in gray).

Our equation turned out to be like this:

The area of the "whole rectangle".
A = (4x - 1)(3x + 5)
A = 12x2 20x - 3x - 5

The area of the smaller rectangle (in gray).
A = (2x - 3)(3x - 1)
A = 6x2 - 2x - 9x + 3
A = 6x2 - 11x + 3

Subtract them together.
A = (12x2 + 17x - 5) - (6x2 - 11x + 3)

Note: To subtract polynomials, change the "-" sign (in the middle) to "+" then change the rest of the right side to it's opposite signs.
A = 12x2 + 17x - 5 + (-6x2) + 11x - 3
A = 6x2 + 28x - 8

Aaaand that's it. Pheew, did I do this blogging right?
Homework: Exercise #1 - If you're not done yet.
Homework: Exercise #3

Did you do it right...are you kidding???? You have set a new standard of excellence in this class. Terrific job recapping todays events and explaining the concepts. Your colour coded visuals really help the reader know what steps to take for distributive property and long division. Way to Go Jeff.

Wow!!

Excellent use of colour, mathematically all correct, well laid out and organized. This is everything a scribe post should be; and it's really helpful for other students to learn from. I feel like I was there ... and I wasn't. ;-)

This post belongs in The Scribe Post Hall Of Fame!

Keep up the good work Jeff. You've set an excellent example for the scribes that follow you.

Darren Kuropatwa
DMCI

WOOOOWWWW~!!!!
nice job. that beat MY post. lol...
mrs. amstrong!! looks like u have a good bunch of students this year.. **still not the beat.. coz i aint there** . ROFL!!! KEEP UP THE TREMENDOUSLY AWESOME WORK!!

CArPE DIeM!!

good job !

oh btw .. how did you make those exponent thingy on the blog ? ms armstrong showed me last year i think but i forgot how , i keep using ^^^

There is a link on the right sidebar that shows you how to format an exponent.

wow, my work's added to the hall of fame, thanks =D

good job!..thats gonna be hard to beat!

At November 16, 2006 5:19 p.m.,  Anonymous

Hi,
Nice visuals, but is the long division correct? You might want to check the actual math.

At January 28, 2007 3:57 p.m.,  Anonymous

alright kiddo...you have WAY too much time on your hands...haha jk...this blog is a GREAT IDEA!!!!

hi mrs amstrong good job..from pelo karuba

At January 29, 2008 6:19 p.m.,  Anonymous

thanks! you've been a great help!

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At September 21, 2010 1:42 p.m.,  Anonymous

Very creative. One minor thing to watch out for here... In the final exercise where you are asked to find the area of the shape, notice that the bottom side equals 3x+5, whereas the "top" sides added together give 3x. Hence, if this is a rectangle, you have 3x= 3x+5 or 0=5 which is impossible. So the error is in the problem, not how the solution is approached. I make similar mistakes all the time when teaching- argh! I usually offer extra credit to any student who can catch a mistake like this.

The way in which you have described the method of solving multiplication and dividing of polynomial is very useful for students.They can easily learn the method from it.