Adding and Subtracting Polynomials

Hi!
Today I was going to post about dividing polynomials, but I figured it'd be better to start with addition and subtraction first- you use them in division, and they're important to know!
Because I hate typing out variables with indices and because drawing pictures on photoshop takes a long time, I decided to do these by hand... So I apologize if you cannot read my handwriting, or if the colors are a little off.
Also- if you are to be wary of any incorrect information in this post, it's to be wary of the positives and negatives. I occasionally have trouble with them... [awkward cough]

But, without further delay...
Addition and subtraction of polynomials!

To add the polynomials, you can lay them out horizontally and add the variables that way. However, I find that a little difficult, and harder than the second way. The second way is to stack the numbers in a vertical way, and then add down.

Of course, this is where those pesky positives and negatives come in. However, it's not that hard to explain- I even made a picture for you about this as well.

Basically, remember that two negatives equal a positive, and that a negative plus a positive means, basically, the positive minus the number that is negative. (So -14 + 2 can also be phrased as 2 - 14, which is also 14 - 2, only with a negative sign on the end. If any of that made sense...)
Just remember your positives and negatives and you'll be fine.

To subtract two polynomials, you basically turn it into an addition problem.
Here's the picture I drew (I'll explain it more fully after the picture.):

Let's describe what's happening in this picture.

First of all, you have your polynomials. I'm going to simplify these and simply do this problem as an example:

(x + 2) - (2x - 8).

If you look at that, what you really have are two different phrases, attached to each other. (Remember, the positive and negative sign attaches- what looks like a minus can actually be used as a negative.)

Those two phrases are:

+ (x + 2)

and

- (2x - 8).

To transform this, you want to change that negative sign in " - (2x - 8)" into a positive sign. To do that, simply multiply the (2x - 8) by -1 (or simply switch the signs.) This will give you:
+ (2x + 8).
Adding those two together will give you
(x + 2) + (2x +8) =
This also works for negatives- let's switch our negatives/positives around for a second...
(x - 2) - (2x + 8).
Now your two phrases are "+ (x - 2)" and "- (2x + 8)". Again, simply transform the negative sign on (2x + 8)...
You'll then have "+2x - 8". And so on and so forth.

Take another look at my picture now that I've explained all the steps in it...

Does that make sense?

So, really, all you need to remember is how to do addition, and how to turn a subtraction problem into an addition one, and you'll be fine! :)

Pie Charts

Time for some statistics, then.

Pie charts.
What are they?
They're circles with petty colors.
No.
Pie charts are useful for demonstrating parts and percentages in relation to the whole. Here's an example of a pie chart (and please excuse the badly drawn circle.)

Note that you can't really take any information from this, because we don't have any labels on here.
Let's change that.

Here's a pie chart now saying what most writers do when they say they're writing (using a graph with a better drawn circle)...

Now this graph, you can get some information out of. Not only can you see that most authors spend way too much time looking at writing websites and social networking, but that, compared with the whole, it's pretty huge.
But how can you tell exactly what these percentages are?
for example...
What if you had this pie chart?

You can still see that the majority of authors spend too much time online- but you can't tell how much, exactly.
So, how do you find that out?
With your protractor!

Measure the angles of the pie chart, and you'll get something like this...

Now we have the angles of the different sections! But how do we figure out anything else, then? With a series of mathematical problems (duh).

To figure out the percentage of these, you simply put the angle over 360 (total angle of a circle). For example, here's one of them worked out.




Take 43.2 and use long division to transform it into .12 (360 goes into 432 one time, remainder of 72. Drop the zero. 360 goes into 720 two imes, no remainder). Then move the decimal place two points to the right, and add a percentage sign.

 Therefore, an angle of 43.2 means 12%. If you do that same thing with the rest of the different angles, you'll get the chart we previously had.
But what if you want to figure out exactly how many people did each?

Well, first of all, you need to either figure out how many is in one sector, OR you need to figure out how many people there are as a total.
Let's use the first one for an example.

Let's say for a minute that the percentage of authors who actually write when they say they're writing (12%) equals 20 authors.

Now that you know that 12% = 20 people, you can figure out what the whole is- and from there you can figure out what the different sectors are!
The easiest way (I've found) to do this is to simply divide the amount of people by the percentage, making it into a fraction, and then multiplying that by 360.
(Easier than it sounds)
Let's take a look at that.

20 and 12 are both divisible by 2. Dividing them both equals 10 over 6. To make 360 into a fraction, you put it over 1. 360 and 6 are both divisible by 6. 6 divided by 6 equals 1 (making 10/6 into 10/1, or an even number of 10), and 360 divided by 6 equals 60 (making 360/1 into 60/1). Therefore, 20/12 x 360/1 = 10 x 60 = 600 = total amount of authors.

Now that we have 600 as the total amount of people in this pie chart, we can figure out what the other sections are.
This leads into the next part.
...How in the world do you figure out what amount of people per sector there are if you have the total amount and the angles of the parts?!
What do you do?!
Okay. Let's look at our pie chart (with the angles) again.

Now, we know that the total amount of people is 600.

But how do we figure out, say, the amount of people who do nothing when attempting to write? (Aka- staring blankly at their computer screen for hours).

First of all, you have to take the angle of the sector and put it over 360. Then multiply that by the total amount. The number you get will tell you how many people per sector there are.

Allow me to demonstrate.



360 and 600 are both divisible by 60. 360 divided by 60 equals 6. 600 divided by 60 equals ten. Therefore 46.8/360 x 600/1 = 46.8/6 x 10/1. 10 and 6 are both divisible by 2. Divide both. You get 46.8/3 x 5/1. Divide 46.8 by 3 using long division. 3 goes into 4 one time, remainder of 1. Bring down the 6. 3 goes into 16, 5 times, with a remainder of 1. Bring down the 8. 3 goes into 18, 3 times, with a remainder of 0. That gives you 15.6 x 5. Multiply them. You get 78 (after multiplication you move the decimal the total amount of places in the two numbers- in this case, one. That turns 780 into 78.0, or 78.) Therefore 46.8/360 x 600/1 = 78.



After that, you can see that the answer is 78- that is, there are 78 authors who, when saying that they're going to write, just sit there and stare at a blank page instead. XD

To draw a pie chart, it's quite easy. Start with a circle.

Draw the radius.

From this, measure your first angle. (In this graph's case, 90 degrees).

Then the next angle (in this case, 12 degrees)(and please note I'm sort of estimating on this... Not really measuring. XD)

Then the next one... and the next one... and the next one... Always measuring from the line right next to the sector.

Ta da! Pie chart. Then you can figure out the percentages, if you want, or just color it, and add your labels... Whatever floats your boat (and satisfies your math exercise/problem... xD).

Simple Area Formulas: Rectangle, Parallelogram, Triangle, Trapezium

I'm aware that most people have already covered/learned these formuals. However, they're incredibly important no matter what level of math you're at, they're easy to learn, and they make a good first math-y post. :)

The formulas, without explanation, are as follows:

Where "A" equals the area of a shape, b equals the length of the base, and h equals the height of the highest side in relation to the bottom edge.

Rectangle: A = b x h

Parallelogram: A = b x h

Triangle: A = ½ x b x h

Trapezium: A = ½ x h (x + y)

In the last formula, for the trapezium, x and y are the parallel sides.

If you just need to figure out the area right away, those are good to use.

But if you're like me, and you need to understand exactly why something works- or at least the general idea- then here are my explanations.

Rectangle: A = b x h

(Area equals base multiplied by height)

Why?

Take a look at my illustration.

Judging from my picture, if you take A = b x h, that's A = 4 x 3. Anyone who has learned simple multiplication will know that that means that the area of this rectangle is 12².

Again, this is an incredibly simple explanation. To understand, simply draw the lines along the measurement markers...

Count the squares (or rectangles, considering my lousy job with my illustrations) and you'll see that there really are twelve squares, leading to yet another way to demonstrate this simple formula.

Parallelogram: A = b x h

(Area equals base multiplied by height)

Why?



And I gave up on trying to make them
look neat... X_X



You may notice that the formula for finding this shape is the exact same as it was for finding the area of the rectangle. The reason for this (which you can possibly see from my sloppy illustration) is because a parallelogram is, essentially, a squashed rectangle. But wait, you might be thinking. That can't be right- after all, it looks more like a square with two triangles shoved on!

Yes, it is, in a way. Here's another illutration showing how we can make it look more like a simple rectangle... By taking triangle YZX and moving it to the other side, we form a perfect rectangle. Divide that along the measurement marks (just like we did with the rectangle) and you can see how this formula works. (Again, the area of this parallelogram is 12², just like it was for the rectangle.)



Triangle: ½ x b x h

(One half multiplied by the base multiplied by the height)

Why?

The reason for this formula is quite simple. Take triangle ABC. Its base is 3 and it has a height of 4. If you add another triangle of the same dimensions (or, considering this is an iscoceles triangle, split the congruent triangle A'B'C' in half) and rearrange it, you get this rectangle. Your original triangle is half of the said rectangle (or square, if your original triangle is an equalatreral) leading to A = b x h (area of your rhombus) divided by half, leading to the area of your triangle.

Tip: When solving it, it's best to align your problem above the 2- especially if you have decimals.

Trapezium: A = ½ x h (x + y)
(Area equals one half multiplied by height multipled times the sum of the two parallel sides)

Why?

First let's take a look at the shape itself.

A trapezium has two parallel lines, with one being shorter than the other. The other two sides are diagonal- although the angles against x and y are not always equal.

When you first look at the formula for finding the area of a trapezium, it can look a little bewildering. But the reasons why you use this formula- instead of finding the area of the rectangle/square in the center, then the area of the two triangles on the sides, then adding them together- really does make sense. The next illustration shows basically what we're doing to this poor trapezium when we use this formula.

Basically, we're taking the triangle from one side of the trapezium, cutting it off, flipping it upside down, and then applying it to the other side of the original shape, which creates a rectangle or a square. We can do this because the two sides (x and y) are parallel, meaning that they're the same difference away from each other on both sides, meaning that the places where the triangles are attached (first trapezium illustration- the dotted line is this place where the triangle is attached) are the same length on both sides.

To be honest, the ½ is applied to the (x + y) section of the formula. Take another lok at our example trapezium.  x + y = 16. 16 x ½ (or 16 divided by two) = 8. Go back up and look at our transformed trapezium. The two parallel sides now equal 8- we have equalized their lengths. Because the distances between the two are constant, we're left with a rectangle.

Because I have time, and because 7, 8, and 9 have so little differences between the numbers, let me show you how this formula works for a trapezium with a greater number difference...

A = ½ x h (x + y)
A = ½ x 3 (2 + 10)
A = ½ x 3 (12)
Multiply ½ times the 12
A = 3 x 6
A = 18

There you go.

Simple formulas. :)

By the way: Notice I didn't include circles here?

That's beause circles have formulas all their own.

Might cover those later. XD

~Angela

About the Title, and this blog:

Hi!

I created this blog because I happen to be a math geek... And enjoy talking about the stuff I geek about. (Aka- math.)
On this blog I'll be posting stuff like... How to solve various problems, various math problems themselves, and other tidbits about math.

The title is actually a math formula... pi r squared is for finding the area of a circle.

I hope you enjoy my blog!