Is T_k a Subspace for Polynomial Coefficients Summing to k?

[supercali]

1. Homework Statement
A) let T_k be all polynomials with degree 3 or under such that k is their coefficients sum.
so we can say that exist at least 2 values of k for them T_k is the sub vector space of P_3(x)?
B) this question is about direct sum: Let $$V_1,V_2,V_3$$ be subvector spaces of V if $$V_1 \cap V_2={0}$$ and $$V_1 \cap V_3={0}$$ than $$V_1 \cap (V_2+V_3)={0}$$

Homework Equations

A) is this true? am i right? look under for my answer

The Attempt at a Solution

A)i think this statement is false for example for k=2,4 we have T_k polynomials but T_k is not close under scalar multipication...is this right?

B)i infact don't think the statement is true but i couldn't find an exaple to support it

 

[Dick]

For A), k=2 or 4 don't work, you are right. But what if k=0? For B), yes, it's false. Take the space of linear polynomials P_1(x). Let V1 be all multiples of (1+x), V2 be all multiples of 1 and V3 be all multiples of x.