FP1 Help

I need some help with this following type of question:

Show that the equation

x^3-12x-7.2

has one positive and two negative roots.

I'm not sure how the exam board (Edexcel) expect as proof. Would a simple finding all three intervals of the roots using the change of sign rule it teaches be enough?

Also, if the question instead asked to prove that something was the largest positive root, how would I go about proving that? Would inspecting it with the change of sign rule to find an interval for the root and then saying that since increasing x would never change the sign of f(x) after the interval given that the interval for the root found is the largest be sufficient or am I missing something about the change of sign rule?

(Since this is part of numerical solutions, I'm assuming it doesn't want me to actually factorize or anything similar to solve)

(edited 12 years ago)

Reply 1

nuodai

There are a few ways to do this.

One is to find places where the graph if positive and negative and use the sign change argument to show that there must be roots in those places. In this case you'd need to show that it's negative for large negative x, that there's somewhere when x is negative where it's positive, that there's somewhere after that (but with x still negative) where it's negative, that it's negative at zero and that it's positive much later on when x is positive. This is a bit cumbersome, but is fine.

A slightly slicker way is to consider the stationary points. It turns out that this function has a derivative which is easy to factorise, so you can find the x-coordinates of the stationary points without too much hassle. Then you can find the y-coordinates of the stationary points, and by considering the sign of the y-intercept (i.e. whether it's positive or negative) and the shape of a cubic graph, you can deduce that it has one positive root and two negative roots.

Reply 2

msmith2512

I agree with nuodai's method and it is by far the most mathematically pleasing but ...

... x large and negative then function large and negative, x=-1 function is positive, x=0 function is negative, x large and positive then function large and positive gives root #1 < -1, -1< root #2 <0 and 0 < root #3. Is much quicker.

Reply 3

nuodai

Original post by msmith2512

That's the first method I mentioned in my post.

Reply 4

msmith2512

Yes it is but you said it was cumbersome. In this case it isn't it is actually the quickest.

Also in my post I did say the stationary points method is the 'correct' way.

Reply 5

nuodai

Original post by msmith2512

Yes it is but you said it was cumbersome. In this case it isn't it is actually the quickest.

Also in my post I did say the stationary points method is the 'correct' way.

I'm not sure if it's worth arguing about this, but I should justify my point for the sake of the OP. The reason why the second method is 'slicker' than the first is because it tells you where to look. The fact that using -1 as your negative point to check was down to luck, and if this wasn't your first guess you'd have needed to keep searching until you found a negative value of

x

for which the function was positive. For example, if I presented you with the equation

y=x^3+23.1x^2+121.1x-145.2

(which has two negative roots and one positive root) then checking large negative

x

and then -1 wouldn't work. In fact, this function is only positive between -12.1 and -12. However, if you differentiate and solve, then you instantly find that it has a stationary point between -12.1 and -12, and plugging this value into the equation tells you that it's positive there (then the other arguments go through).

So yes, if you make a lucky guess then the first method requires fewer calculations. But in general, the second method guarantees a solution, and also illustrates how the geometry of the graph can be used to solve the problem.

Reply 6

ian.slater

Original post by weiyaoli

I need some help with this following type of question:

Show that the equation

x^3-12x-7.2

Looks like this question is in the Edexcel book. Edexcel FP1 will expect you to use change-of-sign. So just find the right values of x to choose. And then it tells you to use Newton-Raphson for the second part.

nuodai's other method is fine, of course. But the FP1 examiner will not expect you to use it.