Posted by Lucy on Sunday, August 31, 2008 at 11:45am.
This is a tricky one. I eventually managed to get an answer out of it, but I have a feeling that the original problem might have been mis-specified (unless of course I've got it wrong myself).
The zeros of this function are easy: all you need are the products of all the factors as follows:
(X+4)(X+3)(X+2)(X+1)(X-2)(X-3)
If you feed in X = -4, -3, -2, -1, +2 or +3 into the above, you'll get f(X)=0 as required, so that's six of the original eleven function values taken care of. However, if you feed the remaining (non-zero) X values into this function, you'll get the following:
f(-5) = 1344 = 48*28;
f(0) = 144 = 48*3;
f(1) = 144 = 48*5;
f(4) = 3360 = 48*70;
f(5) = 18144 = 48*378.
(Just inspecting the five numbers above shows that all of them have a common factor of 48, so we can scale them all to make the numbers a bit easier to handle.)
These figures however aren't the ones we want: what we ACTUALLY want are:
f(-5) = 4; f(0) = 1; f(1) = 1; f(4) = 1; f(5) = 4.
Okay, so let's assume we want to multiply the factorization we got earlier by another polynomial, say g(X), in such a way as to deliver the correct function values. The function values of g(X) will need to be given by:
g(-5) = 4/28 = 1/7;
g(0) = 1/3;
g(1) = 1/5;
g(4) = 1/70;
g(5) = 4/378;
after which we'll have to scale everything by a factor of 48. That is, we're dividing the function values we want by the ones we're actually getting at the moment.
There are five points here, so we can get a perfect fit to these points with a fourth degree polynomial, i.e. a quartic. If you set this up correctly either on your calculator or in a spreadsheet, you'll end up with an equation that goes through all the points. Take that equation, divide it by 48 and multiply it by the factorization for all the zeros that we got earlier. You should find that the resulting product delivers the right answer for every one of your original eleven points.
Having said all that, I reckon it might be an idea to check to see whether you've got the original problem specified correctly!
Correction: f(1) = 240 = 48*5. I got the factorization right, but copied down the original number wrong. Whoops.
I checked and it is copied down correctly.
The answers I have to choose from are:
a) y=0.02x^4-0.25x^2+0.11x+0.84
b) y=0.2x^4+0.25x^2+0.11x+0.84
c) y=0.2x^4-0.25x^2+0.11x+0.84
d) y=0.02x^4+0.25x^2+0.11x+0.84
Thanks for any help you can give.
Ah - you didn't mention that you were selecting the answer from a list of options - that would have saved a lot of time! Having said that, I've just checked all four, and none of them produces the f(X) values listed. Not surprisingly, because the function values listed describe a quite irregular pattern - certainly not something that a quartic would be expected to fit exactly. But then I see now that you're not TRYING to fit the points exactly; just to get the best quartic equation that you can find - in which case the answer is a), as you can verify by just working out the function values for all four of the options available, and comparing them with the list of function values. The easiest way to do that would probably be with a spreadsheet like Excel, if you have access to one; failing that, the Windows calculator will do.
Thanks bunches