The set of 4-degree polynomials (Linear algebra)

[nietzsche]

Homework Statement

If P is the set of all 4-degree polynomials, and W is the subset of all 4-degree polynomials such that p(-2) = p(2), find a set S such that W = span(S).

Homework Equations

The Attempt at a Solution

My guess is that one set that works is x^4, x^2, and 1. My reasoning is that if x^3 is included, then you could take the coefficients of everything to be zero, and (-2)^3 =/= 2^3. Can't include x for similar reasons.

Really not sure, I feel like it is possible to construct a polynomial where p(-2) = p(2) but does include an x^3 or an x.

 

[Dick]

Uh, 'I feel like' doesn't really help. Why don't you try and construct a polynomial such that ax^3+bx=p(x) and p(2)=p(-2)? Since you only have one restriction on the five dimensional subspace P, you really would expect W to have dimension 4.

 

[nietzsche]

ah, it took me a while but i got one.

x^3 + 3x^2 - 4x - 12

i guess i don't really understand the question then. how could such a set be constructed?

 

[Dick]

ah, it took me a while but i got one.

x^3 + 3x^2 - 4x - 12

i guess i don't really understand the question then. how could such a set be constructed?

It took too long because you were working too hard. You already know the span of 1, x^2 and x^4 are in W. The question you asked, and suspected is true, is whether a combination of x and x^3 are in W. If p(x)=ax^3+bx, then p(2)=8a+2b., p(-2)=(-8a-2b). The condition for them to be equal is 4a+b=0, right? So right again, you should include x^3-4x in the spanning vectors. You didn't need to fuss with the even power terms. You already know they work.

 

[nietzsche]

oh, i see. it makes perfect sense now. oh well, i guess i will just get part marks for that question then... thanks for the explanation.

 

[HallsofIvy]

A general fourth degree polynomial is of the form p(x)= ax^4+ bx^3+ cx^2+ dx+ e. Then the condition that p(2)= p(-2) is that p(2)= 16a+ 8b+ 4c+ 2d+ e= p(-2)= 16a- 8b+ 4c- 2d+ e. a, c, and e cancel immediately leaving 8b+ 2d= -8b- 2dor 16b+ 4d= 0. That reduces to d= -4b while a, c, and e can be anything. Taking a= 1, b= c= d= e= 0 gives x^4. Taking b= 1, a= c= e= 0, d= -4 gives x^3- 4x. Taking c= 1, a= b= d= e= 0 gives x^2. Taking e= 1, a= b= c= d= 0 gives 1. A basis for this space is {x^4, x^3- 4x, x^2, 1}.