Finding a Polynomial with Nonnegative Coefficients

In summary, to find a polynomial P(x) with nonnegative coefficients, given that P(1)=6 and P(5)=426, we can start by setting a0=1 and using the conditions to find the other coefficients. We can then use the equation 4a1 + 24a2 + 124a3 = 420 to determine the values for a1, a2, and a3. From there, we can continue to find the remaining coefficients and determine the value of P(3).
  • #1
chillfactor
12
0

Homework Statement



find a polynomial P(x) which has nonnegative coefficients. If P(1)=1 and P(5)= 426, then wast is p(3)

Homework Equations


P(1)= 6
P(5)= 426
P(3)= x

The Attempt at a Solution


I have tried to use guess and check. I can't find a way to solve algebraically.
 
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  • #2
Do you have a specified degree for p(x) ?
 
  • #3
There is no specified degree for p(X). It just has to have nonnegative coefficents
 
  • #4
chillfactor said:
There is no specified degree for p(X). It just has to have nonnegative coefficents

Then it is best to choose a degree and work with that.
 
  • #5
There is only one polynomial that works. One hint I received was that it had to be less than 6.
 
  • #6
I have no idea how to approach it either. Can't quite gather what to do with the fact that all the coefficients aren't negative. Either way, it must be at least a polynomial of degree 4. But what suggests that it must be less than 6?
 
  • #7
how do you know it must be at least degree 4?
 
  • #8
If you try to fit for example a quadratic

[tex]ax^2+bx+c[/tex] to the conditions, then you get two conditions:

a+b+c=1

25a+5b+c=426

We can subtract one from the other to get

24a+4b=425

Now we know that a+b+c=1 must be satisfied, with all non-negative numbers, so none of a, b or c can be larger than 1. That's clearly not possible if 24a+4b=425. A similar argument kills cubic polynomials
 
  • #9
i am afraid i made a mistake when I posted the question. Actually P(1)= 6. So, could you try again. I would appreciate it.
 
  • #10
There are lots of choices of polynomials actually, just consider a degree n polynomial of the form axn+b. We know that a+b=6, and that 5na+b=426. Two equations, two unknowns, start trying to find solutions!

If you know a little linear algebra/convex geometry there's a neat explanation for how you can decide exactly which coefficients are able to be non-zero
 
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  • #11
welcome to pf!

hi chillfactor! welcome to pf! :wink:
chillfactor said:
find a polynomial P(x) which has nonnegative coefficients. If P(1)=6 and P(5)= 426, then wast is p(3)

i assume all the coefficients must be whole numbers?

then ∑an = 6 and ∑an5n = 426

obviously a0 = 1,

and by subtracting we have 4a1 + 24a2 + 124a3 = 420 …

carry on from there :smile:
 
  • #12


thanks!
 

FAQ: Finding a Polynomial with Nonnegative Coefficients

What is a polynomial with nonnegative coefficients?

A polynomial with nonnegative coefficients is a mathematical expression that consists of variables, coefficients, and exponentials, where the coefficients are all positive numbers. For example, 5x^2 + 3x + 2 is a polynomial with nonnegative coefficients, but -2x^3 + 4x - 1 is not.

Why is finding a polynomial with nonnegative coefficients important?

Finding a polynomial with nonnegative coefficients is important because it allows us to solve certain problems in real-world applications, such as optimization and modeling. It also helps us to simplify mathematical expressions and make them easier to work with.

How do you find a polynomial with nonnegative coefficients?

To find a polynomial with nonnegative coefficients, we can use a mathematical technique called the method of undetermined coefficients. This involves setting up a system of equations and solving for the coefficients that will result in all positive values. Another approach is to use polynomial interpolation, where we can choose the points we want the polynomial to pass through and find the coefficients that satisfy those conditions.

What are some common applications of polynomials with nonnegative coefficients?

Polynomials with nonnegative coefficients have many practical applications, such as in finance, economics, and engineering. They can be used to model and analyze data, optimize processes and systems, and solve various mathematical problems. They are also commonly used in computer science and data analysis.

Can a polynomial with nonnegative coefficients have negative solutions?

No, a polynomial with nonnegative coefficients can only have positive or zero solutions. This is because the coefficients represent the values of the polynomial at different powers of the variable, and negative powers would result in negative values. However, a polynomial with nonnegative coefficients can have complex solutions, which include both real and imaginary parts.

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