- [Instructor] So we have the graph here
of Y is equal to P of x,
I could write it like this, Y is equal to P of x.
And they say, what is the remainder
when P of x is divided by x plus three?
So pause this video and see if you can have a go at this
and they tell us your answer should be an integer.
So, as you might have assumed,
this will involve the polynomial remainder theorem
and all that tells us is that hey if we were to take P of x
and divide it by x plus three,
whatever the remainder is here,
so we'll say the remainder is equal to k.
That value k is what we would have gotten
if we took our polynomial
and we evaluated it at the value of x
that would have made x plus three equal zero
or just what would have happened
if I evaluated our polynomial as x equals negative three.
You have to be very careful there
sometimes people get confused.
They see a positive three
and then they evaluate the polynomial at the positive three
to figure out the remainder.
No!
If you saw a positive three there,
you would evaluate the polynomial at negative three
but this should be equal to k as well.
And so what is the remainder when P of x
is divided by x plus three?
Well it's going to be equal to P of negative three.
P of negative three it looks like
it is equal to negative two.
It is equal to negative two, so our remainder
is equal to negative two in this situation.
Let's do another example,
actually let's do several more examples.
Here we're told that P of x is equal to all of this business
where k is an unknown integer, very interesting.
P of x divided by x minus two has a remainder of one.
What is the value of k?
So pause this video again, see if you can work it out.
All right, well this second sentence
that P of x divided by x minus two has a remainder of one,
that tells us that P not of negative two
but P of positive two,
whatever x value would make this expression equal zero.
That P of two is equal to one.
And then we could use this top information
to figure out what P of two would be.
It would be two to the fourth power
minus two times two to the third power
plus k times two squared.
So, times two squared minus 11,
and so all of that, that's P of two right over here
that's going to be equal to one.
Two to the fourth is 16
and then two times two to the third
that's two to the fourth again, so it's minus 16
plus four k minus 11 is equal to one, these cancel out.
Now let's see we can add 11 to both sides of this equation
and we get four k is equal to 12.
Divide both sides by four and we get k is equal to three
and we're done.
Let's do another example, in fact let's do two more
because we're having so much fun.
So this next question tells us, P of x is a polynomial
and they tell us what P of x divide by various things are,
what the remainder would be when you divide P of x
by these various expressions.
Find the following values of P of x,
P of negative four and P of one.
Pause this video and see if you can have a go at it.
All right, so P of negative four,
this is going to be equal to the remainder
when P of x divided by what.
You might be tempted to say x minus four
but they're trying to trick you intentionally.
This would be the remainder when P of x
is divided by x plus four.
And so they tell us right over here
P of x divided by x plus four has a remainder of three.
So, it's going to be three right over there.
And similarly P of one, this is going to be the remainder,
this is the remainder when P of x divided by
not x plus one, but x minus one.
So when P of x is divided by x minus one,
the remainder is zero.
Let's do one last example.
So once again P of x is a polynomial
and then they give us a few values of P of x.
And they say what is the remainder
when P of x is divided by x minus three?
Pause the video and try to think about that.
Well, we've gone over this multiple times,
the remainder when P of x is divided by x minus three
that would be P of not negative three, P of positive three.
Whatever value of x makes this entire expression equal zero.
So P of positive three is equal to five.
And similarly what is the remainder,
actually not so similarly, this is interesting.
What is the remainder when P of x is divided by x?
I know what you're thinking
it's like wait what number am I dealing with?
But if I were to rewrite this
instead of saying divided by x,
if I were to say divided by x plus zero
then you'd be like oh now I get it.
Or if I wrote divided by x minus zero,
you'd be like oh, now I get it.
This is going to be P of and it doesn't matter
whether I take a positive or a negative zero,
it's going to be P of zero.
And P of zero, they tell us, is negative one and we're done.