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Objective

Review over Adding and Subtracting Polynomials
Review over Multiplying Polynomials
Go over the Fibonacci Sequence
Go over it’s importance (once you see it, you can’t unsee it)
Go over Pascal’s Triangle
Go over the Binomial Theorem
Do some examples
Homework

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So, what is a monomial?

Basically, a monomial is a one termed portion of an equation.

It’s commonly referred to as simply a term, and the most important thing about them is that we combine like terms.

�We’ve actually been doing this for quiet some time, but here is an example to refresh your memory:

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Adding Monomials

Imagine you are given something like:

You’ve probably heard that when you see something like this, you need to combine like terms.

What that means is, every portion, or monomial, in this equation, needs to be added to its like component.

So, just for practice, let’s list out all of the different terms:

But that’s not all right? We also have each of their like components (or monomials)

So now, let’s add them together and see what we get!

And now, we are left with:�0 + x – 12 = 0�

Or�

x – 12 = 0�

+ 12 + 12

x = 12

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And that’s pretty much it

That is essentially how we add monomials together, we add like terms.

Now, things do get a little more complicated than this to begin with, but for the most part, it’s all the same.

So, let’s do a few more examples, without listing the monomials, to make sure you get it.

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Example 2

Add the following:

2xy + 3x – 2y + 4xy – 2x + 4y

So, again, we want to add like terms, which may seem tough since there are 3 different ones, but we can do it.

So, to start with, let’s add:

2xy

4xy

2xy + 4xy

= 6xy

Now, let’s add:

– 2x

3x – 2x

= x

And finally, we add:

– 2y

+ 4y

– 2y + 4y

= 2y

And we are left with:

6xy + x + 2y

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SO, HOW CAN WE TELL DIFFERENT MONOMIALS APART?

Sometimes you may get a little confused about what is and is not a like term, and that’s okay!

But, to understand them completely, so you can do add them correctly, let’s look at what the monomial is actually is.

So, if we look at our example of:

2xy

What does this mean?

Well, remember, anytime you see a number, next to things that aren’t a number, it’s multiplication.

So, in this case, this actually means:

2 * x * y

However, we’re lazy here in the math world, so we just skip the *, and keep it as: 2xy.

So what’s the difference between 2xy and 2yx?

Well, what’s the difference between 2*3*4 and 2*4*3?

Nothing!

And that’s the exact same with monomials! �

Now, you shouldn’t get any monomials like that in your

homework, but it still wouldn’t hurt to go over it.

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Example 3

Add the following:

2y + 3x + 3yx – y – 2xy – 3x

Again, we need to add the like terms.

And don’t forget, 3yx is the same as 3xy.

So let’s rearrange so we can add them easier:

2y – y + 3x – 3x + 3yx – 2xy

Now let’s isolate them so it’s a little easier:

(2y – y) + (3x – 3x) + (3yx – 2xy)

y + 0 + xy

y + xy

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Multiplying Polynomials

So we know how to add and subtract polynomials by simply combining like terms.

But how do we multiply polynomials?

Well, to be honest, it’s not as hard as it may sound.

Basically, we make sure to multiply each monomial from one polynomial to all of the other monomials of the other polynomial.

Wow, that sounds way more complicated than it is.

Here’s an example:

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Example

Multiply the following:

To multiply these all together, we multiply each part of the first polynomial by each part of the second polynomial.

Like so:

___________________________________________________________________________

So, this is the product of what we started with.

Again, we need to make sure we multiply each part of the first polynomial

With each part of the second polynomial.

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And that’s how we do it!

So, that’s how we multiply polynomials together.

We multiply the smallest polynomial by the biggest polynomial.

Now let’s go over the Fibonacci Sequence

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A FEW DEFINITIONS TO GO OVER

Recursive – something is recursive if you do it repeatedly.

Sequence – is a list of numbers.

So a recursive sequence is a list of numbers that has some sort of repetition. This is a common occurrence in math.

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A few examples of recursive sequences

-5, -1, 3, 7, 11, 15, 19, 23, 27, …..

100, 90, 80, 70, 60, 50, 40, 30, 20, 10,….

4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40,….

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SO THERE’S PATTERNS IN RECURSIVE SEQUENCES

Even though it isn’t technically part of the definition of a recursive sequence, it is very common to see a pattern appear from recursive sequences.

For example, in the first example on the previous slide:

-5, -1, 3, 7, 11, 15, 19, 23, 27, …..

We can see that the number is getting 4 added to it. We know this because if we take any number (other than -5) and subtract the number before it, it will equal 4.

Example:

7 – 3 = 4, -1 –(-5) = -1 + 5 = 4, 11 – 7 = 4, etc.

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So why does this matter?

Sequences are found everywhere, whether its in Algebra, calculus, trigonometry, geometry.

It would also be irresponsible to have a powerpoint about sequences and not include the most famous one of them all…..

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The Fibonacci Sequence

The Fibonacci sequence was founded by the Italian mathematician Fibonacci

Fibonacci wanted to know, given a certain amount of time, how many rabbits you could farm if you started with only two baby rabbits.

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Big deal right?

Well, it is a big deal. Fibonacci wrote all of the numbers down that he found, and this is the sequence he came up with:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc.

So can you guess what the sequence is?

….

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But that’s not all!

See, Fibonacci was nuts and decided he wanted to make a geometric shape in response to his findings.

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SO HE STARTED WITH A SQUARE WITH AN AREA OF 1

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So what? Some weirdo found a curve with boxes.

Big deal right? Except this is a

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See, the Fibonacci spiral can be found in:

In your DNA

The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.

�

Your fingerprint, which rotates in the same way

Your ear and it’s creation while you were in the womb

The symmetry of your face

A snail shell

Roses

Succulents

Wood Shavings

Seahorses

Pinecones

Cakes

Ocean Waves

Snake Coils

Hurricanes

Artichokes

Jupiter’s Storm (which is as big as Earth)

Our Solar System

The Galaxy

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BUT THAT’S NOT ALL!

Believe it or not, there is even music that follows the Fibonacci Sequence. Listen:

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Alright, we get it

The Fibonacci Sequence is super important, but for Algebra 2 (at least for what we’re going over right now) what Fibonacci’s Sequence gave rise to was:

Pascal’s Triangle.

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This is Pascal’s Triangle, who used the Fibonacci sequence to help us understand binomials.

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SO, THAT’S HOW THE BINOMIAL THEOREM WORKS

By realizing that binomials multiply in the same way that Pascal’s triangle can be added, we can figure out what the answer is without actually doing the multiplication.

So, that is the Binomial theorem and how it works.