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
Mentor
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
Homework Helper
Gold Member
2020 Award
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
Homework Helper
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