Name: 代數基礎。簡化多項式 - 數學反思錄 (Algebra Basics: Simplifying Polynomials - Math Antics)
Uploaded: 2021-01-14T06:24:00.000Z
Duration: 10 min 43 s
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In our last basic algebra video, we learned about polynomials.

Specifically, we learned that polynomials are chains of terms that are either added or subtracted together.

And we learned that the terms in a polynomial each have a number part and a variable part that are multiplied together.

If you don’t remember much about polynomials, you might want to re-watch the first video before you continue.

But even though the basics of polynomials are pretty simple,

sometimes you’ll come across polynomials that are more complicated than they really need to be.

And in math, what do we like to do when things are too complicated?

So in this video, we’re going to learn how to simplify polynomials.

Simplifying a polynomial involves identifying terms that are similar enough

that they can be combined into a single term to make the polynomial shorter.

To see how that works, have a look at this basic polynomial that follows an easy to recognize pattern.

Of course, as I mentioned in the last video,

we don’t really need to show the coefficients of each term if they are just ‘1’ like we have here.

And the ‘x to the zero’ term is also just ‘1’, so we don’t really need to show that either.

But I’m going to leave it like this for just a minute to illustrate my point.

As you can see, this polynomial has a term of every degree from zero up to four.

But, do you remember that it was okay for a polynomial to have “missing” terms?

For example, we could have a slightly different polynomial that doesn’t have a third degree term.

That makes it look like the ‘x cubed’ term gets skipped or is missing,

since the pattern goes from ‘x to the fourth’, then skips ‘x cubed’ and goes to ‘x squared’ and so on.

Well, just like there can be missing terms in a polynomial,

there can also be EXTRA terms… like in this polynomial, where the third degree term has been duplicated.

See how there are TWO terms that have an ‘x cubed’ variable part in this polynomial?

So THIS polynomial has NO ‘x cubed’ term (which is fine)

and THIS polynomial has just ONE ‘x cubed’ term (which is fine)

but THIS polynomial has TWO ‘x cubed’ terms (which is also fine)… BUT…

it’s more complicated than it needs to be!

terms that have the exact same variable part…

To do that, you just add the number parts and you keep the variable part the same.

So, one ‘x cubed’ plus one ‘x cubed’ combine to form two ‘x cubed’.

What we just did there is called “Combining Like Terms”.

‘Like’ terms are terms that have exactly the same variable part.

Well to understand that, I like to pretend that the variable parts of a polynomial’s terms are fruit.

For example, have a look at this polynomial.

But let’s substitute a different kind of fruit for each different variable part.

If we do that, what would this new fruit polynomial be telling us?

Well, this first term represents 2 apples,

So that raises the question… what do you get when you add 2 apples to 4 oranges?

Well… you get… 2 apples and 4 oranges!

Since they are different fruit, you can’t combine them.

Ahh… but what about the middle two term?

What do we get if we add 4 oranges and 3 oranges?

And that means we CAN combine these two terms into a single term which makes our fruit polynomial simpler.

Now do you see why the variable parts of a term have to be exactly the same in order to combine them?

If the variable parts are different (like ‘x cubed’ and ‘x squared’)

then they represent different things, so we can’t group them into a single term

the way that we can if the variable parts are the same.

The mathematical reason that it works that way has to do with something called The Distributive Property,

which is the subject of a whole other video.

Alright, so if two terms in a polynomial have exactly the same variable part,

then we call them ‘like’ terms and we can combine them into a single term to simplify the polynomial.

And to help you get better at identifying ‘like’ terms,

let’s play a little game called “Like terms or NOT like terms?”

The first pair of terms we’ll consider is 2x and 3x.

Yep!  The variable part of both terms is the same (x) so we can combine them into a single term.

We do that by adding the number parts and keeping the variable part the same.

Well… they’re both first degree terms, but since the variables are different letters, they are NOT ‘like’ terms.

Two ‘x squared’ and negative seven ‘x squared’.

Well the variable part in both is exactly the same. It’s ‘x squared’.

So YES, these are ‘like’ terms and we can combine them.

Notice that one of terms is negative, so when we add the number parts we’ll end up with negative 5.

So these combine to negative five ‘x squared’.

Our next pair of terms is four ‘x squared’ and six ‘x cubed’.

Nope!  Even tough the variable is ‘x’ in both cases, the exponents are different, so the variable parts are not the same!

Well, at first glance, you might think that the variable parts of these terms are different

because the ‘x’ and the ‘y’ are in a different order.

But remember, multiplication has the commutative property so the order doesn’t matter.

so we can re-write them so they look the same too.

negative 5 plus 8 is 3. So we wind up with the single term 3xy.

Last we have five ‘x squared y’ and five ‘y squared x’.

You might think that it’s like the last one where the terms are just in a different order, but look closely.

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代數基礎。簡化多項式 - 數學反思錄 (Algebra Basics: Simplifying Polynomials - Math Antics)

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