Adding and Subtracting Polynomials
Objective
THE DOMAIN OF A FUNCTION
Now that we understand how to represent an interval on a Number line, let’s talk about Domain.
So, for the official definition, the Domain of a Function, is all of the possible values of x.
Now, for some graphs, there is a limit on what x can be, but those are usually stated before hand.
For most other’s, there is no limit, because the function is unbounded.
For example:
If we look only at the x-axis of this graph, we can see that this graph has no limit on what x can be.
Therefore, we would say that:
Domain:
Inequality:
Set Notation:
Interval Notation:
However, if we are given a graph such as:
We would then see that:
THE RANGE OF A FUNCTION
Now that we understand how to identify the domain of a function, let’s talk about Range.
So, for the official definition, the Range of a Function, is all of the possible values of y.
Now, for some graphs, there is a limit on what y can be, but those are usually stated before hand.
However, it is important to note, they are more common than having a limited domain.
For most other’s, like with domain, there is no limit, because the function is unbounded.
For example, looking at our past example:
If we look only at the y-axis of this graph, we can see that y must be greater than or equal to zero.
Therefore, we would say that:
Range:
Inequality:
Set Notation:
Interval Notation:
However, if we are given a graph such as:
We would then see that:
LOCAL MAXIMUM AND LOCAL MINIMUM OF A FUNCTION
A local maximum is a maximum that can be seen given a specific interval.
Like the graph we saw before, some graphs have what we call valleys (or low points) and peaks (or high points).
These highs and lows are considered the local maximums and local minimums of a function.
The peaks are the maximums, and the valleys are the minimum.
So why do we call them local?
Because they are specific to the interval given.
The actual maximum of a function is usually referred to as the global maximum, and is the highest point of the function at all intervals. �Or, in other words, the highest point of the entire function.
So let’s look at a quick example to tell the difference.
THE LOCAL MAXIMUM
Let’s use the same graph (because I’m lazy and don’t want to go making another one).
Again, it’s important to point out that the local maximum is not the highest point of the entire graph, it’s just the highest point of that interval.
THE LOCAL MINIMUM
Now let’s talk about the local minimum.
The local minimum is the same as the local maximum, except that it’s the lowest point of the graph.
So, take the same graph (again, lazy!).
The zeroes of a function
Finally, the zeroes of a function are where the graph crosses the x-axis.
This only occurs when you set y = 0.
They are also fairly easy to spot, so let’s look at an example:
As we can see from this graph, at (0,0) and at (3,0) it crosses/touches the x-axis.
So the zeroes of this graph would be: x = 0, x = 3.
And yes, that’s how you would write that.
End behavior
Last thing we’ll talk about concerning graph review (at least for right now) is their end behavior.
Basically, the end behavior is what the graph will do constantly in either the positive direction, or the negative direction.
The way we determine what the end behavior of a graph is, is by determining the direction we want to face, then determine what the graph does.
This is more complicated to explain than show, so here’s an example:
Example:
Determine the end behavior from the following function:
As we can see, as x increases, we can see that this graph seems
to fall to negative infinity.
So, to answer this question, we would say:
Likewise, if we want to look at as x decreases, we can say that
the graph seems to rise to positive infinity.
So, we would also say:
DOMAIN, RANGE, AND END BEHAVIOR OF DIFFERENT POLYNOMIALS
So, to start, let’s look at different graphs to see if we can find some similarities:
So, as we can see, even degree and odd degree polynomials follow a pattern with their domain and range, as well as their end behavior.
DOMAIN, RANGE, AND END BEHAVIOR OF NEGATIVE POLYNOMIALS
So, to start, let’s look at different graphs to see if we can find some similarities:
So, as we can see, even degree and odd degree polynomials follow a pattern with their domain and range, as well as their end behavior even if their leading coefficient is negative.
TURNING POINTS OF POLYNOMIAL FUNCTIONS
So, the turning points of polynomials is where the polynomial changes from going in the positive direction, to the negative direction, or vice versa.
For example, with the graph:
We can see the graph turns at around x = -2
We can see the graph turns again at around x = 0
So the turning points are:�x = -2, x = 0
FINDING THE X - INTERCEPTS
So, to find the x-intercepts of a polynomial, you set the polynomial equal to 0, and solve.
Whatever x equals, those are the x-intercepts of the function.
For example:
Example
Find the x-intercepts from the following:
f(x) = x(x - 7)(x + 7)
So, again, to find the x-intercepts, set the function equal to 0, then solve.
So:
0 = x(x - 7)(x + 7)
Now, before we lose it, let’s think about this.
We have three parts of this function being multiplied together.
But, if only 1 of those parts ends up being 0, then the whole function is 0.
So, we can set each part to 0, and get the answers we want.
So:
x
x - 7
x + 7
= 0
= 0
= 0
+7 +7
-7 -7
x = 7
x = -7
So then x = 0, 7, -7
So, now that we’ve reviewed, 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:
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
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.
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:
3x
– 2x
3x – 2x
= x
And finally, we add:
– 2y
+ 4y
– 2y + 4y
= 2y
And we are left with:
6xy + x + 2y
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.
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