Finding a Basis for the Subset of Polynomials Satisfying p(5)=0 | Exam Prep

I found the first polynomial by letting b= 1, c= 0 in 25c+ 5b+ a= 0 and the second by letting b= 0, c= 1 in 25c+ 5b+ a= 0.In summary, the set U is a subset of the vector space V, where all members satisfy the equation p(5) = 0. The subspace U has dimension 2 and can be spanned by the basis {1, x-5, x-5}, where x is a real number. This can be determined by setting the parameters b and c equal to 1, 0 and 0, 1 and finding two linearly
  • #1
tsMore
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I am drawing some strange mental blank with one question in my final exam review.

Homework Statement



V is the set of all polynomials that are of the form p(x) = cx^2 + bx+a
U is a subset of V where all members satisfy the equation p(5) =0

Find a basis for U.

I am not sure why I am having so much of a problem with this one, it shouldn't be that hard. With 5 subed in you get polynomials of the form

25x^2+5x+a = 0

I guess I am having a hard time relating this a back to a basis. Isn't the basis for V just {x^2, x, 1)?

Anyway thanks in advance for all your help!
 
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  • #2
Wrong. With 5 subbed in you get 25*c+5*b+a=0. That's a relation between the coefficients - not all polynomials satisfy it. The subspace has dimension 2 (why?), so can you find two linearly independent polynomials that do?
 
  • #3
Polynomials satisfying p(5) =0 on the form p(x) = cx^2 + bx+a can be expressed as factors: p(x) = A(x-5)(x+B) ; where A and B is a real number.

Now how about that?

Edit: Dick was 5 seconds before me on this one too =P
 
  • #4
malawi_glenn said:
Polynomials satisfying p(5) =0 on the form p(x) = cx^2 + bx+a can be expressed as factors: p(x) = A(x-5)(x+B) ; where A and B is a real number.

Now how about that?

Edit: Dick was 5 seconds before me on this one too =P

Ok, I'll close my eyes and count to 5 before I answer the next one.
 
  • #5
Dick said:
Wrong. With 5 subbed in you get 25*c+5*b+a=0. That's a relation between the coefficients - not all polynomials satisfy it. The subspace has dimension 2 (why?), so can you find two linearly independent polynomials that do?


Wow that was fast :) And you are probably hitting on exactly why I am confused as well. Guess I am am just phasing out with this.

I actually don't know why the subspace would need only two polynomials to span the space defined by p(5) =0. I was actually thinking it was three!
How did you determine that it was 2?

You are saying that 25*c+5*b+a=0 is the relation between the coefficients, but not all the polynomials for which p(5)=0 need to statisfy this. What am I missing there? I was trying to relate the coefficents together some how to come up with a definition of the space.

thanks again so much!
 
  • #6
All of the polynomials such that p(5)=0 DO satisfy 25*c+5*b+a=0! Name one that doesn't. V is spanned by your basis {1,x,x^2}, dimension 3. U is a subspace, it has smaller dimension. I can easily think of two polynomials that span it. I agree that you may be phasing out. Step out and get a breath of fresh air and take another look at the problem.
 
  • #7
The vector space P2, of all quadratic polynomials is 3 dimensional and is spanned by {x2, x, 1}. Any member of that space is of the form cx2+ bx+ a. The subspace of such polynomials for which f(5)= 0 must satisfy 25c+ 5b+ a= 0. All that has been said before.

From 25c+ 5b+ a= 0, you can get a= -25c-5b. In other words, all polynomials in that subspace are of the form (-25c- 5b)x2+ bx+ c.

One method I really like for finding a basis for such a subspace is to take the parameters (here b and c) equal to 1, 0 and 0, 1 successively. If b= 1 and c= 0, then (-25c- 5b)x2+ bx+ c is simply -5x1+ 5x and then, if b=0 and c= 1, -25x2+ 1. Those two polynomials form a basis for the 2 dimensional subspace of quadratic polynomials f(x), such that f(5)= 0.
 

1. What is a subset of polynomials?

A subset of polynomials is a group of polynomials that is a smaller part of a larger set of polynomials. It contains only a specific selection of polynomials that meet certain criteria.

2. What does p(5)=0 mean?

The notation p(5)=0 means that the polynomial p, when the variable is replaced with the value 5, results in an output of 0. In other words, the polynomial has a root or zero at x=5.

3. How do you find a basis for a subset of polynomials?

To find a basis for a subset of polynomials satisfying p(5)=0, you can use the method of finding a null space. This involves setting up a matrix with the coefficients of the polynomials and using row reduction to find the linearly independent polynomials that form the basis.

4. Can a subset of polynomials have more than one basis?

Yes, a subset of polynomials can have multiple bases. This is because there are different ways to choose linearly independent polynomials that satisfy the criteria. However, all bases for a given subset will have the same number of elements, known as the dimension of the subset.

5. Why is finding a basis for a subset of polynomials important?

Finding a basis for a subset of polynomials allows us to represent any polynomial in the subset as a linear combination of the basis polynomials. This can help simplify calculations and make it easier to understand the properties of the subset. It is also important in applications such as solving differential equations and interpolation.

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