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What happens if you're graphing a factorised polynomial function
and some of the factors appear more complex than you would like?
Well, the simple fact is that it's not FULLY factorised!
Now that may not be a problem but let's look at an example.
Here's a simple quadratic where we have two factors multiplied together.
This is an extremely simple factor. This one's more complicated.
And I hope you can see (I've chosen a very simple example)
that we can take a common factor of five out of this.
Now, I hope you don't mind —
instead of putting it in front of this factor here,
I'm going to take it right to the front and we get this factorization
where 5 times x gives 5x and 5 times -2 gives -10.
Now we can see, again, that if we want to find the roots of this quadratic,
setting y equal to zero, that x would be -2 or plus 2.
And if I was to graph this —
there's a root at +2 and a root at -2, as we've discovered.
Notice, there's no minus sign so we know that,
as the x values get large, the y values get quite large as well,
so we know the graph goes this way.
What effect does the five have? This is what the variation is all about.
What does a number, a leading coefficient of 5 or 10 or 15 or 18 do to a polynomial?
Well, the simple fact is that it means that the y value gets much, much larger.
Whatever this product is worth, we have to multiply it by five.
And it has the effect of making the graph very much steeper.
For a small increase in x we get a very much larger increase in y.
So, this parabola, in fact (and this is going to disappear from your video),
would go way down below the axis.
What would be the effect of having a SMALL factor [leading coefficient]?
Well, let's check that out.
What if we had something like 1/2(x-2)(x+4)?
Again, finding the roots, there'd be a root at x=2 and a root at -4.
Let's plot the roots – two, negative four.
It's a positive value, so we know the graph is rising as x increases,
but it's only a half.
It means that, whatever value this expression takes on,
we halve it before we find the y value.
And that has the effect of "shrinking" the y-axis.
So, instead of being a relatively steep parabola,
it now is rather flattened. (I don't think I did a good job of that.)
So, leading coefficients are worth studying because they tell you the steepness of a graph.
Let's apply it [the concept] to a straight line.
I'm going to choose a fairly simple line,
and we can consider this as a polynomial with one factor.
Notice, the coefficient of x is 2 and we can remove that as a common factor.
Now, 2 into 5 goes 2 1/2 times (or 2.5 times) so, 2 times x is 2x and 2 times -2 1/2 is -5,
and this is the factorization that we would like.
Again, to find the root of this polynomial we set y=0
and we find there's a root when x is worth 2 1/2.
So, as you can see, another consequence of having more complicated factors is
sometimes the roots are not whole numbers. So, we plot this root at 2 1/2.
We know that it's a positive polynomial (or, at least, the leading coefficient is positive)
which means that the polynomial will increase as x increases, and "2" means it's steeper.
So, instead of going out like this it actually is steeper.
Now, in the case of lines, it's very simple.
It means for every value of x you go across you go up two squares.
And you can see that, in fact, the distance along the x-axis
is going to be half the distance along the y-axis.
So, if that's 2 1/2, this is going to be at -5.
And, of course, that's the y-intercept that we can read from the equation anyway.
But, this is just a slightly different way of looking at the equation for a straight line —
to identify its root by factorising.
We're going to do one last example, and that's a slightly more complicated one,
to put it all together and show that we can have
a negative, large leading coefficient with roots that are not all whole numbers —
so, let's have a look at that.
Now, let's graph just one more polynomial.
We're going to draw one that has a simple factor, one that has a reversed factor,
and one that has a slightly more complicated factor.
And let's see the effect of Putting all this together.
If we fully factorise this, the first term doesn't need any change
but [in] the second term, the coefficient of x is -1
and we can take that out (outside the parentheses)
so that minus x is -x and minus -5 gives plus 5.
And here, the coefficient of x is 3, and we can take that outside the parentheses
and fully factorise this factor.
Three times x is 3x.
And, you'll excuse me if I use fractions rather than decimals,
but 3 times 2 2/3 makes 8.
We can keep these factors together but these two constants, the -1 and the 3,
we can multiply and bring out the front – and straight away you can see the effect
of having reversed terms and more complicated terms.
That is, we sometimes get more complicated roots and we do get a leading term.
If we expanded this we get x times x times x would give us the x-cubed
and its coefficient would be -3 — and there would be other terms.
What is the effect of that -3?
Well, I hope you know by now, but let's draw this graph.
To find its roots we set y=0. We can then determine the three roots.
There'll be a root at +2, a root at +5 and a root at -2 2/3.
We draw the axes, one root at 2, another root at 5 (... -2 and -3 ...),
so -2 2/3 would be about there and our graph has to pass through
these three roots, these three zeros, these three points.
The leading coefficient of -3 means —
the minus sign means that the graph must head downwards at the right
(as x increases in value, y will increase negatively),
and the 3 means it will do so steeply.
So, instead of going in a nice smooth [I meant 'gentle gradient'] curve
this will go down quite steeply which means, of course,
it will rise quite steeply here.
Now, I'm going to disappear off the bottom of the video here.
This will go way, way down below and up through here
and you can see our cubic equation — a typical "S" shape.
But, there it is, the effect of the first two variations on our theme,
having reversed and more complicated factors.
If you're very fast, you can sometimes just inspect the terms
and read that we have a root at 2, and a root at 5 ('cause 5-5 makes zero) —
if you can calculate quickly, you can see that we have a root at -8/3 which is -2 2/3.
And then if you check the leading coefficient by looking at your x values,
we can multiply 1 by -1 by 3 which gives us the -3 and an understanding of what's going on.
So, you need not do all these calculations if you're particularly fast at calculating.
Now I encourage you to watch the third (and last) of my three videos
describing those three variations on the theme of how to graph factorised polynomial functions
using their roots.
In my third video I'll be explaining what happens when a root is repeated.
So far, we haven't seen that,
and I'll be explaining that there are certain principles involved
that make the graphing of those functions very simple as well.
Thank you for watching.