Is S a Subspace of P_3 and Does q(x) Belong in S?

[dragonxhell]

Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :(

Let S be the following subset of the vector space P_3 of all real polynomials p of degree at most 3:

S={p∈ P_3 p(1)=0, p' (1)=0}

where p' is the derivative of p.

a)Determine whether S is a subspace of $P_3$
b) determine whether the polynomial q(x)= x-2x^2 +x^3 is an element of S

Attempt:
I know that for the first part I need to proof that it's none empty, closed under addition and multiplication right?
will this give me full mark for the part a if I answer like this:
(af+bg)(1)=af(1)+bg(1)=0+0=0 and
(af+bg)′(1)=af′(1)+bg′(1)=0+0=0
so therefore it's a subspace of P_3?
b) i got no idea...

Thank you very much!

 

[Mark44]

Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :(

Let S be the following subset of the vector space P_3 of all real polynomials p of degree at most 3:

S={p∈ P_3 p(1)=0, p' (1)=0}

where p' is the derivative of p.

a)Determine whether S is a subspace of $P_3$
b) determine whether the polynomial q(x)= x-2x^2 +x^3 is an element of S

Attempt:
I know that for the first part I need to proof that it's none empty, closed under addition and multiplication right?
will this give me full mark for the part a if I answer like this:
(af+bg)(1)=af(1)+bg(1)=0+0=0 and
(af+bg)′(1)=af′(1)+bg′(1)=0+0=0
so therefore it's a subspace of P_3?

What allows you to say that (af+bg)(1) = 0 and that (af+bg)'(1) = 0? You haven't used the fact that S is a subset of P₃. You also haven't shown that the zero function belongs to S.

b) i got no idea...

The set description tells you which functions belong to S. Namely, they are of degree less than or equal to 3, p(1) = 0, and p'(1) = 0. Does q satisfy all three of these conditions? If so, it's in S.

 

[PeroK]

What you have for a) is correct. You need to say what f and g are to write it out properly.

What is stopping you from checking whether q(x) is in S?

 

[pasmith]

Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :(

Let S be the following subset of the vector space P_3 of all real polynomials p of degree at most 3:

S={p∈ P_3 p(1)=0, p' (1)=0}

where p' is the derivative of p.

a)Determine whether S is a subspace of $P_3$
b) determine whether the polynomial q(x)= x-2x^2 +x^3 is an element of S

Your question is identical to the question asked in this thread.