# Homework Help: Impossible pre cal question Give formula for the graph of the polynominal

1. Jul 3, 2012

### xdrgnh

1. The problem statement, all variables and given/known data

http://www.math.poly.edu/courses/ma0914/past_exams/MA0922_Final_2000-12-13.pdf [Broken] First problem on the exam

2. Relevant equations

I believe the answer should be a 3rd degree polynomial so Ax^3+Bx^2+Cx+D

3. The attempt at a solution

Ax^3+Cx
-8A-2C=0
8A+2C=0

-A-C=3
-4A=C
-A+4A=3A

A=1.

However that is impossible because A has to be negative for the 2nd derivative on the left side of the equation to be positive. Idk where I went wrong.

Last edited by a moderator: May 6, 2017
2. Jul 3, 2012

### DeIdeal

This is not correct. It should be a 5th degree polynomial. Are you able to explain why & continue from there?

There are several possible ways to form the necessary equations to find out the constants.

3. Jul 3, 2012

### xdrgnh

I guess I would continue by plugging in the values for the zeros and plug in the values for (-1,3) and (1,-3) and solve the system of linear equations. However I don't know why it's a 5 degree. Technically I'm not supposed to use derivatives.

4. Jul 3, 2012

### DeIdeal

Something like that, yeah : ]

I'm not too familiar with the US (I assume?) education system, so I don't know what is taught in Pre-calculus. If, however, you already know how to take the derivative of polynomials, you might want to use that in order to make the system of equations easier to solve. It shouldn't make a huge difference, though.

As for why it's 5th degree: You should be able to see it from the shape of the graph. One way to explain it is that a 3rd degree polynomial can only change direction two times (ie. f'(x) has only two roots and is a 2nd degree polynomial). You can see that the graph changes from increasing to decreasing or vice versa four times, so f'(x) has four roots and f(x) is a 5th-degree polynomial. Well, at least that's one possibility, and you were only looking for ONE possible formula.

5. Jul 3, 2012

### xdrgnh

Alright I got it. I Went from Algebra 2 and trig to calc 1 and 2 and then all of the way up to complex variables. However to get a job as a math tutor I need to take a test in pre calculus. However contrary to the name pre calculus I noticed I never used any of this stuff in my upper math classes.

6. Jul 3, 2012

### xdrgnh

-32A+16B-8C+4D-2E=0
32A+16B+8C+4D+2E=0
-A+B-C+D-E=3
A+B+C+D=-3

So I get 4 equations and 5 unknowns. However when I try to solve them I get.
2B+2D=0
32B+8D=0
which implies that both B and D are zero does that seem right so far.

7. Jul 3, 2012

### DeIdeal

Yeah, B=0, D=0.

EDIT: You should get 5 equations for the 5 unknowns, though.

Well, six for the six unknowns, but f(0)=0 -> F=0 is pretty damn obvious, so.

8. Jul 3, 2012

### xdrgnh

Where would the 5th one come from besides the face that F would equal zero. Because I'm getting something that can have infinitely many solutions.

9. Jul 3, 2012

### DeIdeal

You can use f'(2)=0 or f'(-2)=0, for example. There's an exact solution, don't worry.

10. Jul 3, 2012

### xdrgnh

I can't use derivatives this pre calc. Is this question possible without calculus?

11. Jul 3, 2012

### xdrgnh

Also wouldn't the 2nd derivative be zero not the first at 2. After looking I noticed that the 1st derivative is zero also. Not like it helps.

12. Jul 3, 2012

### DeIdeal

Oh. To be honest, I didn't really think about that possibility. Using derivatives just seemed so obvious that I didn't bother to check how to do it without them, sorry.

I'm not sure about this, but I don't see any way to do this without the derivatives.

EDIT: No, the first derivative is zero at the (local) maxima and minima. The sign of the second derivative tells whether it's a maximum or a minimum, unless f''(x)=0.

13. Jul 3, 2012

### xdrgnh

Well even though my original hunch that it's not possible is correct I see that my reasoning was wrong. Thanks for the help I learned something.

14. Jul 3, 2012

### Filip Larsen

It is a bit easier if you are familiar with the fundamental theorem of algebra, from which it follows that if a n-degree polynomial has the roots (or zeros) x1, x2, ..., xn (some of which may be equal to each other), then the polynomial can be written as g(x) = a(x-x1)(x-x2)...(x-xn), where a root will be repeated if the function has zero derivative (i.e. horizontal tangent) in that root.

Looking at the graph you can count 3 roots, of which 2 is repeated since they have a horizontal tangent, and 4 horizontal tangents in total. Taken together that means g must be a 5-degree polynomial with the root in -2 and 2 each repeated twice. From the last information, that g(-1) = 3, you can then solve for a and get one particular polynomial that "fits" all the information given.

15. Jul 3, 2012

### xdrgnh

I wrote down the g(-1)=3 and got 4 equations with 5 unknowns.

16. Jul 3, 2012

### D H

Staff Emeritus
The graph appears to be <fill in the blank> about x=0. Use that.

17. Jul 3, 2012

### xdrgnh

symmetric?

18. Jul 3, 2012

### D H

Staff Emeritus
Close. There is a symmetry there.

Hint: How do f(x) and f(-x) appear to be related?

19. Jul 3, 2012

### klondike

f(-x)=-f(x) which allow us setting all even degree (include f) to zero. Makes life easier but still need the knowledge of derivative. How this can be done without derivative?

20. Jul 3, 2012

### skiller

Filip Larsen (post #14) has suggested all you need to know to solve it. There's no need to bother about whether it's "symmetric" or looking at any derivatives or any simultaneous equations (as the first few posts were suggesting).

21. Jul 3, 2012

### D H

Staff Emeritus
Well there is a need for symmetry considerations if you read the zero at x = -2 as a zero at "?" In other words, specifying where that zero occurs is part of the problem.

OTOH, if one reads the graph correctly, then post #14 says it all. No calculus needed.

22. Jul 3, 2012

### skiller

EDIT:

:smilie:

Last edited: Jul 3, 2012
23. Jul 3, 2012

### Curious3141

This question can be solved by just observation, and minimal calculation.

Whenever a polynomial curve "bounces" tangentially off the x-axis, it means there's a repeated root there. So there are repeated roots at x = -2 and x = 2. Meaning that (x-2)^2 and (x+2)^2 are both factors of the polynomial.

The curve also passes through the origin, meaning that f(0) = 0. Hence x is also a factor.

So you know f(x) can be of the form kx(x-2)2(x+2)2

You're also given f(-1) = 3, so put that into the above equation to work out k.

Last edited by a moderator: May 6, 2017
24. Jul 3, 2012

### SammyS

Staff Emeritus
It looks as though there is a lot of traffic on this thread, with an occasional visit by OP.

25. Jul 4, 2012

### xdrgnh

Well guys I believe I got it via following post 14 advice. -(1 / 3) * x * (x + 2) ^ 2 * (x – 2) ^ 2. I remember learning about the fundamental theorem of Algebra in my junior year of high school but because I never used it in my physics and future math classes I forgot it. I got my pre caluclus test this Thursday and this Wednesday I'm going to be studying for it. Do you guys have good online resources. The problem with these test questions is that they have no answers to them.