Let's looks at polynomial functions
of the variable x.
Of course, the first thing we need to know
is what is a polynomial function,
and what is not.
So, let's look at some examples.
All of these are polynomial functions
of the variable x.
Notice that all of the x's are raised to a
non-negative integer power.
Square roots of numbers are okay, that's
just a real number, but we want all the x's
to be raised to a
non-negative integer power.
None of these are polynomials.
The problem with the first one is there's an
x in the denominator.
The problem with the second one is there's
an x under the square root, so that's not allowed.
That's not a polynomial.
This one has a negative exponent on the x,
also not allowed, and this one has a
fractional exponent on the variable x which
is not allowed for a polynomial.
For these that are polynomials that we can
talk about the degree of the polynomial
function of the variable x.
What we would like to look at are the terms,
and we want it multiplied out, not factored.
So we want to pick out the exponent on the
term that is the highest,
the largest exponent.
Alright so for this one, the highest
exponent I see is 2 on a x,
so the degree is 2.
This one, the highest or largest exponent I
see on x is a 1, so it has degree 1.
And here I just want to be careful I'm not
fooled: the largest exponent occurs
on the last term.
The largest exponent is 5,
so this one has degree 5.
Now let's see about
other things about polynomials.
We'd like to sketch a reasonable graph of a
polynomial function, and there are several
characteristics that will help us to do this.
Let's look at a specific polynomial.
Notice this one is factored
very nicely for us.
The first thing that I'd like to find is the
y-intercept, and we know that, to find a
y-intercept, we let x equal 0 and
evaluate the function.
So I want f of 0.
That's going to be 2 times 0 squared times
negative 4 times 2 cubed, and of course
this is equal to 0.
So, the first thing I know is I have a
y-intercept at 0.
Let's put that on our graph.
The second thing I'd like to talk about are
the x-intercepts and the behavior the graph
does at each one.
I'm going to make a chart to keep track of
all the important information.
Here I want to list my x-intercepts.
In the center I want to
list the multiplicity.
The multiplicity of an x-intercept is the
exponent on the factor from which you get
that x-intercept.
So, we'll talk about that more
in just a moment.
The last column is going to be
for the behavior.
Okay, so the x-intercepts are the values
that would make this equal to 0, values of
x, so certainly 0, positive 4, and negative 2.
Okay, so the 0 comes from the factor x
squared which has an exponent of 2, so the
multiplicity of zero is 2.
The 4 comes from the factor x minus 4
which is to the first power,
so the multiplicity is 1.
And the negative 2 comes from x plus 2 cubed, the
multiplicity must be 3.
Now the behavior is either the graph
will touch or cross the x-axis,
and that depends on the multiplicity.
Where the multiplicity is an even number,
the graph will touch the x-axis without
crossing, and where the multiplicity is an
odd number, then the graph will cross
through the x-axis.
Let's put these x-intercepts on our graph.
The x-intercept at 0, we already have that,
the x-intercept at 4, and then the last
x-intercept is at negative 2.
Okay, the next thing we need to find out is
what happens on the ends of the graph.
So what is the power function the graph
resembles for large absolute value of x?
So we need to figure out, if we multiplied
out the polynomial function, what would the
term with the highest exponent look like?
It would certainly have a factor of two,
it would have an x,
well we have this x squared.
Now let's see what else.
So from this term, we're going to be
multiplying the 2 x squared times a positive
x to the first, and then, from this one, the
highest exponent I can get would be
positive x to the third.
And when we multiply like bases, we add the
exponents so the power function the graph
resembles for large absolute value of x is
2 x to the sixth.
Okay, 2 x to the sixth, this resembles
positive x squared on the ends since the
exponent is even and this coefficient is
positive, so I can imagine the ends of the
graph look like this, they both point up.
Okay, one last question before we finish our
graph: the maximum number of turning points
for this function.
This is always the degree minus one.
Now we know the degree because it also
happens to be the same exponent that's used
on our power function.
So, 6 minus 1 is 5.
I should point out that the graph may not
have five turning points, but it won't have
more than five.
Okay, so how can we get a graph?
Well, the end behavior tells me that on the
left of the leftmost x-intercept, the
function values are doing
something like this.
In my chart I know that when I get to negative 2,
the graph should cross the x-axis,
and it comes down.
Now, I really don't know how far down it
goes, I would really need calculus
to figure that out.
But I do know at some point it comes down
here and it turns and it comes back up to my
next x-intercept.
That one is at 0.
The graph should touch at 0 without
crossing, and then it comes back down.
And again, I don't know how far down because
I'd need calculus, but I do know eventually
it has to turn because it has to hit the
next x-intercept, which is 4,
and the graph should cross there.
Notice that the graph I've drawn is a smooth
continuous curve, no sharp corners, no gaps,
just a smooth continuous curve.
Now I have one last question to answer.
By looking at the graph, I can determine the
intervals of x where the function f is above
the x-axis and those on
which is below the x-axis.
So, if we start on the left, the graph is
clearly above the x-axis from negative
infinity to x equals negative 2, actually just close
to negative 2, so we'll use parentheses on this to
show that the negative 2 is not included.
Right between negative 2 and 0, the graph is below
the x-axis, so I'll list that here.
Now the graph is still below the x-axis
between 0 and 4, so we'll add that down
here, it's below the x-axis.
And then from 4 to infinity, the graph is
above the x-axis again.
Notice that this is two separate intervals:
that's because at zero the graph touches, is
on, the x-axis, so I should not include it.
Let's look at another example.
In this case, we're going to be given the
graph of a polynomial function, and we need
to choose a reasonable polynomial function
from the list, something that
might have this graph.
Okay, so what kinds of things
can we look at?
We can look at the end behavior, and we can
look at the x-intercepts and whether the
graph touches or crosses at each one.
So, let's start with this one.
This x-intercept is at negative 2, and I see that
the graph of the polynomial
crosses the x-axis there.
So, I need to find, in my possible answer, a
factor of x plus 2 to an odd power.
So, right away I can eliminate choice a
because that's an even power, and I can
eliminate choice b because the x plus 2 is
to an even power again.
However, this is x plus 2 to an odd power,
so maybe this works, but so is this one
since it's to the first, and here the x plus
2 is cubed again, so any of these
are still possible.
Let's move on to another x-intercept.
I have an x-intercept here at 0 and I notice
that the graph touches the x-axis without
crossing, and that means I need to find
an x to an even power.
Alright, well if we look at the three
possibilities that are left, they're all the
x's to an even power so I haven't
eliminated another one.
Move on to the other x-intercept at 1, and
again the graph touches without crossing, so
I need to see a factor of x minus 1 that's
raised to an even power.
And I have that here, I have that here, this
isn't x minus 1 to an even power, but it
still gives me an x-intercept at 1 with an
even exponent so it's still
going to touch there.
So, I haven't eliminated anything else.
The next thing I want to look at would be
the end behavior.
Okay, so on the left, I see the end behavior
is something like this, and on the right,
it looks like this.
I want to try to think about what power
function would have this sort of behavior.
Well the power function is definitely going
to have an odd exponent, so maybe x cubed,
we can think about that.
And that's close but x cubed actually looks
like this, so what I have has been reflected
across the x-axis, which means that it's
going to be a negative x to some odd power.
Okay, so I'll see what we have.
For choice c, I want to look for the term
with the highest exponent, so I need to
imagine what I'd get if I can
multiply it all out.
So for c, I'd have 3 x to the fourth, here
the highest power of x I could get
would be x cubed.
Notice it's a positive x there, this is a
positive x and it would be squared, that
would be the highest exponent.
So that's 3 x to the ninth.
It's got the odd exponent but not
the negative coefficient.
Choice c has been eliminated.
Okay, now let's look at d.
I have a negative x squared
to start out with.
And if I were to multiply this out, I would
multiply it by the x there, and over here
this is a positive x to the second power.
Well let's see, that's negative x to the fifth.
That one is good, that's one possibility.
And then notice that more than one answer
might be possible, so I don't want
to give up yet.
I need to look at letter e.
Okay, so negative 5 x squared, if I multiplied this
out, this would give me a x cubed.
And here's where I have to be careful,
the highest exponent would be the negative x squared.
Alright, so overall this is
negative 5 x to the seventh.
Choice e is also an acceptable answer.