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Graph drawing is a wonderful skill,
because it allows us to see a picture of the algebra that we’re studying.
Now, unfortunately,
sometimes it can be quite difficult to draw a graph of an algebraic equation —
but there are some very useful skills that we can learn to make the job more easy.
And the skill I want to explain to you today is how to find the roots
(or the x-intercepts) of a graph.
That is, on our Cartesian graph (or our rectilinear coordinate system)
if we have any sort of graph, we are going to learn today
how to find these points known as x-intercepts
(because the graph intercepts the x-axis at those three points)
and commonly called the roots of the equation.
Now, there are many kinds of graphs that we study in school.
In Australian schools,
by the time a student reaches about fifteen years of age,
there’re expected to know six basic graph types,
that is, straight lines, parabolas, cubic equations, circles, hyperbolae and exponentials.
In their senior two years of school they also learn to graph
trigonometric equations and logarithmic equations.
But, to make our job simple today I’m going to specifically study polynomials
and, even more specifically, FACTORISED polynomials.
Polynomials produce [graphs] like straight lines, parabolas and cubic equations —
the first three [graph types] that I mentioned.
Now, a factorised polynomial looks like this.
If it had one term, it would be a straight line.
With two terms it becomes a parabola, shaped like this.
And if we had a third term it would start to take on an “S” shape —
and we’ll talk about that in another video as well.
But, for the moment, we’re just going to discuss this simple quadratic equation
and how to find its roots.
In order to understand the principle upon which this skill is based
we have to understand that, on a graph, the equation for the x-axis is y = 0.
Not x = 0. It’s the y value that remains zero.
The x value, of course, changes. It can be zero, one, or x = 2, or x = 3, x = -1.
But the y value has not changed because we have not gone up or down the y axis at all.
The y value is zero for the entire length of the x-axis.
Consequently, to find where THIS equation meets the line with THIS equation,
we have to solve these two equations simultaneously,
and their y values must be identical at the points where they cross.
So, we know that if y equals THIS for this equation,
it must also equal ZERO because of the other equation.
And therein lies the secret to finding the roots.
Because, I want you to notice that we have two terms multiplying to give zero.
Now, just take a little while to think of two numbers that multiply to make zero.
I don’t think it will take you very long to realise that it might be something like
0×2 or 0×21 or -4×0 or sqrt(17)×0 — but one of the numbers must always be zero.
You simply cannot multiply two non-zero numbers and end up with zero as an answer.
That means that, if this product between this factor and this factor is to make zero,
then one of those must be zero. So, either (x–2) is zero or (x+4) must be zero.
You have no [other] option. One of those must be true.
Now, if x–2 is zero, we can solve that quite quickly and find that x is worth 2.
If we solve THIS equation we find that, to make that true, x has to be -4.
And these are what we call the roots of this equation
because they make it true.
If x is worth 2 we get 2-2=0 and we don’t care what THAT is worth
because zero times something is zero.
And if x is -4 we get -4+4=0 and again, we don’t care what the other term is
(because anything times zero will be zero).
So these two are very important points.
And if we locate them on the graph where x is 2 and where x is -4,
this graph must pass through those two points.
It turns out that it does THIS — and forms what we know as a parabola.
Now, let’s go over this process again with a different polynomial
and we’ll speed the process up a little and set it out just a little bit more differently.
This time I’m going to graph a more complex polynomial: (x-4)(x+2)(x-3)(x+1)(x).
Here we have five factors all multiplied together and we’re going to find the roots.
Now we know that, on our graph, the equation for the x-axis is y equals nought, or zero,
so we set y=0.
And then we work along the equation, along the polynomial
looking at each term and finding the relevant root.
Here x could be equal to 4 (because 4-4=0); it could be equal to -2 (because -2+2=0);
or here, 3; -1; and, in the case of this term (where x is by itself),
x just has to equal zero in its own right because that makes x=0.
So, here we have five roots for this equation.
And to graph it, we set up our very simple graph paper and we plot the roots.
There’s one at 4, there’s one at -2, another one at +3, another one at -1, and one at 0.
There are our five roots.
Now the nice thing that I didn’t explain earlier about [graphs of] polynomial equations
is that they are continuous.
That is, you can draw them without taking your pen off the paper (or the whiteboard).
And this graph must go down through this point, up through this one, down through that one,
up through that one, and down there. And there’s our graph.
It is wonderfully simple to do.
Now, let me do it one more time and show you how I would set it out in an examination paper.
Let’s imagine that we’ve been asked to graph y = (x-3)(x+5)(x-1).
I would write a heading.
I would write the heading ROOTS to explain that’s exactly what I’m finding,
and I would explain how I’m going to find [them], that is, I’m going to set y=0.
I like to underline my headings.
And I would write y = (x-3)(x+5)(x-1)=0.
I would write my roots.
I like to line my “equals signs” up.
My roots would be at 3 (because, again, 3-3=0), at -5 and at +1.
And then (I’m a little cramped for space here) I would draw a graph.
I would mark on the roots: +3, at -5, and at +1, and draw my graph.
Through there, up there, and down there. And there it is!
As quick and as simple as that! And I encourage you to practice the skill.
I hope you agree that this is a simple process.
In my next video I’m going to explain
what happens if one or more of these terms is repeated.
I’m also going to explain what happens if a term is reversed,
that is, if we had 1-x instead of x-1.
And I’ll also explain what happens
if the terms are just a little bit more complex than this.
In the video following that one, I’m then going to solve a number
of past HSC exam [questions] from NSW senior school examinations
so you can see how these skills are applied to competitive examination questions.
In the meantime,
I hope this helps you find the roots of factorised polynomials in a very simple way
and that you can graph them and even find enjoyment in graphing them.
I hope you also realise why it is so important
that you learn how to factorise polynomials efficiently.
This is Graeme Henderson wishing you the best with your mathematics.
See you in the next video.