# Thread: Showing a vector subspace and finding its dimension

1. ## Showing a vector subspace and finding its dimension

Again, do not have any idea where to start with this:

'Show that the subset of R[x]n (n should be subscript here), (n≥4), of those polynomials whose graph touch the x-axis tangentially at x=0 and x=1 is a vector subspace and find its dimensions. Also, find a basis for this subspace (If you cannot do the case of general n, try n= 4,5 first.)

2. Saying that a polynomial has graph tangent to the x-axis at x= 0 and x= 1 means it has x= 0 and x= 1 are multi-roots of the polynomial and so it is of the form $x^i(x- 1)^j p(x)$ where i and j are larger than 1. (That is why there is the condition that " $n\ge 4$".) Since p(x) is a general polynomial, it can be written in the form $p(x)= a_0+ a_1x+ a_2x^2+ \cdot\cdot\cdot+ a_ka^k$ where k= n- i- j.