When graphing a polynomial that has been factorised,
what does it mean if one of the factors is repeated
(or even, if more than one factor is repeated)?
That'­s what I want to explain in this video —
where I discuss the third variation on my theme of how to use the roots of polynomials
in order to graph them.
Let'­s choose as an example y = (x-2) squared times (x+1).
You can see that I could have written this as (x-2)(x-2)(x+1),
but this is the abbreviated form, using the square.
Now, what effect does this square have on the graph (because it has no effect on the root)?
When we set y=0 we still only get a root at 2 — and we get a root at -1.
They'­re the only numbers that make this expression zero.
Why the square?
Well, this root occurs twice — because this factor occurs twice!
We call it a double root.
And what effect does that have on the graph? Let me explain.
We first need to understand that every graph that is of the form x-squared,
or x-to-the-four, or x-to-the six, or x-to-the-eight
(what we call even functions) looks like a parabola.
It is symmetrical about the y-axis and, as the power increases,
the graph gets steeper and steeper.
We also need to understand that if we have y=x (that is, x-to-the-power-one),
or x-cubed, or x-to-the-fifth, or x-to-the-seventh —
we have what are called odd functions and these graphs come down to zero
and they level off "instantaneously," and they look like an "S" shape.
And, in fact, if you look at them upside down
(if you could stand on your head or turn the monitor upside down),
you would see [that] it would be the same shape when rotated 180 degrees.
It turns out that every factor that has an even power
behaves like THIS near the root;
and every factor that has an odd power (like this has got a power of one)
behaves a bit like this near the root.
The principle to understand here is that, when we'­re examining
the behaviour of a polynomial near one of its roots,
the factor containing that root is dominant.
If for example, we were studying values near 2, that is, near the double root of 2,
this term [factor] is extraordinarily close to zero.
And if I alter the x value by a tiny margin,
x-2 could be as small as a thousandth or a millionth.
When I square it, it becomes even smaller still.
And a slight adjustment in that value can make a huge percentage difference
to this term [factor].
Whereas, if I'­m near 2, this term remains very, very close to three.
Whether I make it 3 plus a millionth or 3 plus a thousandth
is not going to make a big difference to this term,
but it makes a massive difference to THIS one!
Also, if I move to this factor here, when x is near the root -1, THIS dominates.
Because, if I put -1 into here, I get -1 minus 2 is -3 and square it and get nine —
and a small variation, from -1 to -1.01 for example, will not make a big difference.
It'­ll be a number very, very close to nine but it will make a huge difference to this term
(to this factor).
So, the principle is that when you'­re near a root, the factor containing the root is dominant
and that means that this will behave like an even powered function
whereas here, this will behave like an odd powered function.
And if I was to graph it (I'­ll just show you this very, very quickly),
I would still draw my axes, identify the two roots — 1, 2 —
if you'­re doing this in pencil,
it'­s sometimes nice just to write a little "2" there to remind you that it'­s a double root.
There'­s another root at -1 and my graph goes through those two points
even though it'­s a cubic (I have an x-[squared] times x).
And I expect it to be an "S" shape.
The leading coefficient is positive, that is, it'­s got a plus x and a plus x,
so I know the graph goes up here but, as it tries to go through this root at 2,
because this is dominating, it'­s behaving like a square —
very close to this it behaves like a parabola and bends and "bounces off."
It'­s almost as though, because that root is there twice,
it tries to go through it once, and through it again
and never quite makes it through.
And then, of course, the graph must go through the other root.
So, this is the slight variation on the theme that I wanted to show you.
Let'­s try it on a slightly more complicated polynomial.
I'­m just going to make one up now.
Let'­s have an (x-1), an (x+4), an (x-5), an (x), and an (x+2).
I'­m keeping my factors very simple.
And let'­s have a double here, and a triple, and another triple, and another double.
Or, let'­s make this 4 — just to be a little excessive.
Now this is way more difficult than anything that you'­re going to graph at school
or, possibly, even at university (by the way of polynomials).
But, it'­s an x-squared times x-cubed times x times x-cubed times x-to-the-fourth
which means, if I expand it, that I'­m going to have 4, 7, 8, 11, 13 —
an x-to-the-power-13 graph.
But let'­s look at how simple it is to graph using these skills.
Because I need room for the graph, I'­m not going to write the roots out.
We'­ll just work them out as we go.
We have a root at 1 and it'­s going to be a double root,
so I'­ll write a small 2 there.
We have a root at -4 and it'­s a triple root, so I'­ll write a little 3 just to remind me.
I have a root at +5 and it'­s a normal, single root.
I have a root at zero and it is a triple. (These are going to be a bit close together.)
And I have a root at -2 and it'­s a fourth powered root; it'­s there four times.
Now, the leading coefficient is positive (all the "x"s are positive)
so I know that my graph goes up to the right.
It comes down through 5 Because 5 is a norm— (laughs) —
in fact it is the only normal root we have.
When it tries to come up through 1,
being a double root, it behaves like a parabola and it "bounces" down.
It tries to come up through zero but, being a triple root,
it has that little "S" shape of a cubic equation.
It tries to come down through -2 but, being an even powered root,
it bounces off like a parabola, then it wants to come down through the next root
and, being a triple root, it behaves like that odd function again
and it'­s got a little bend in it. And, there we go!
Now, of course, we need to use calculus
to find these maxima and minima and points of inflexion,
but the roots help us get a general picture of the shape of the polynomial very, very quickly.
That'­s a complicated version.
We'­ll do it one more time with a (simpler) cubic.
Now, I'­m going to write the cubic equation up,
but I'­m going to add a slight twist to it as you'­ll see.
Can you see what I'­ve done?
Not only do I have a double root, but this [factor] is reversed.
Notice that the leading coefficient of x is negative.
So, if I rearrange this equation a little, I get the following:
because minus x gives me -x and minus -4 gives me +4.
So, here you can see that I actually have a negative leading coefficient
for the x-cubed and I have two roots.
I can find them by setting y=0. So, x is -2 and x is 4.
They'­re the two possible roots.
And, to graph them, we have one root at -2
(it is a double root, so we'­ll put a little 2 above it);
I have another root at 4 (it is a normal root);
but the minus sign means that my graph starts on the right hand side by going down.
It comes up through the normal root at 4 [and] wants to come down through the root at -2
but, being a double root, it behaves like a parabola.
And there is the shape of that particular cubic equation.
Now, I'­m going to create some work sheets for the original, factored polynomials
that I described about three videos ago,
and other work sheets for the three different variations
that I'­ve described to you in the last two videos (and this one).
Those work sheets will be posted on my website and will be downloadable
from the links shown in the description below this video — in a few days'­ time.
In the meantime, I hope this has helped you understand the principles behind
using the roots of polynomial equations to graph them
(at least, to get the general shape of them).
Thank you for watching.