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## Geometrical meaning of the zeroes of a polynomial

# Polynomial factors and graphs — Harder example

## Video transcript

- [Instructor] We're asked
which of the following could be the equation of the graph above. So pause this video and see
if you can work through this before we do it together. All right, so first of all, let's just think about where the graph intersects the X axis. We can see it does that
at X equals negative four. It does that at X equals zero. And it does that at X
equals positive three. So we would expect to see an expression that's equal to zero
at these three places. So first of all, X equals zero. You could see if X equals zero. All of these will become zero, 'cause zero times anything else is zero. So they all meet that one. See, X equals negative four. In order for it to become zero
at X equals negative four, we'd want to see an X plus four when we factor things out. We see an X plus four here, we see an X plus four here. These second two, they have X minus four. So we don't like those. So we can immediately rule those out. And then for X equals three to make the entire
expression equal to zero, you'd need to have an X minus three. And we can see we have an X minus three and an X minus three. So now we have to decide
what makes more sense to have a negative X out
here or a negative X squared. Well, let's think about what would happen in either case as X increases. As X increases, as X
becomes larger and larger positive values, you take
the negative of that, and it makes sense in this A choice that the value would get
more and more negative. That's also the case in choice B. As X increases, you square it. You'll get even a more positive value. And then you take the negative of that, it should go down. So both choice A and B is
consistent with this behavior of as X becomes more positive, Y is becoming more and more negative. Now let's look on the other hand. What about when X becomes
more and more negative? Well, when X becomes more and
more negative in choice A, when you take the negative of a negative you would expect this part to
become more and more positive. So you would expect for choice A, that actually the Y values would increase as X becomes more and more negative, which clearly isn't happening. So we can rule that one out, as well. But let's just verify that B works. So when X becomes more and more negative, you take the square of that. So you're going to get a positive value, but then you take the negative of that. So it's going to become,
so Y will keep decreasing and becoming more and more negative, which is exactly what
we see in this graph. So we're liking choice B.