TheWrathOfMath
Junior Member
- Joined
- Mar 31, 2022
- Messages
- 162
V=P4[x] (the subspace of all polynomials of degree 4 or less).
Find the basis of the polynomial subspace W, which includes all polynomials in V such as that when multiplied by (x-2), the product will be an odd degree polynomial.
I only managed to write:
---------------------------?
W={p(x}∈V | p(x)*(x-2) = Σ an*x^(2n+1)
--------------------------n=0
Looking at the p(x)*(x-2) part:
Two questions:
1) Is it preferable to write the sigma notation in words instead, since there is no numerical upper limit for the "largest odd number"?
2) How do I proceed to find the basis? Perhaps I have done something wrong or went about this in a complicated manner for no good reason?
Find the basis of the polynomial subspace W, which includes all polynomials in V such as that when multiplied by (x-2), the product will be an odd degree polynomial.
I only managed to write:
---------------------------?
W={p(x}∈V | p(x)*(x-2) = Σ an*x^(2n+1)
--------------------------n=0
Looking at the p(x)*(x-2) part:
(a+bx+cx^2+dx^3+ex^4)*(x-2) =
= ax+bx^2+cx^3+dx^4+ex^5-2a-2bx-2cx^2-2dx^3-2ex^4 =
= -2a+(a-2b)x+(b-2c)x^2+(c-2d)x^3+(d-2e)x^4+ex^5
.Now, I know that the above expression must be a polynomial with an odd degree.
The condition is met in the following cases:e=/=0
e=0 and c-2d=/=0
e=0 and c-2d=0 and a-2b=/=0
Two questions:
1) Is it preferable to write the sigma notation in words instead, since there is no numerical upper limit for the "largest odd number"?
2) How do I proceed to find the basis? Perhaps I have done something wrong or went about this in a complicated manner for no good reason?