Subspaces in Polynomial P_5(x) of Degree < 5

In summary, the question is asking which of the following subsets of P_{5}(x) are subspaces: (i) the set of all polynomials in P_{5}(x) of even degree, (ii) the set of all polynomials in P_{5}(x) of degree 3, (iii) the set of all polynomials p(x) in P_{5}(x) such that p(x)=0, and (iv) the set of all polynomials p(x) in P_{5}(x) such that p(x)=0 has at least one real root. To determine if these subsets are subspaces, we need to test if they are closed under linear combinations. For (i)
  • #1
gtfitzpatrick
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If P[tex]_{5}[/tex](x) is the set of all polynomials in x in degree less than 5. Which of following subsets of P[tex]_{5}[/tex](x) are subspaces.
(i) the set of all polynomials in P[tex]_{5}[/tex](x) of even degree
(ii) the set of all polynomials in P[tex]_{5}[/tex](x) of degree 3
(iii) the set of all polynomials p(x) in P[tex]_{5}[/tex](x) such that p(x)=0
(iv) the set of all polynomials p(x) in P[tex]_{5}[/tex](x) such that p(x)=0 has at least one real root

i'm really not sure but this is what i think
polynomial =ax[tex]^{4}[/tex]+bx[tex]^{3}[/tex]+cx[tex]^{2}[/tex]+dx+e
(i)im not sure about the question but i think it means

a1x[tex]^{4}[/tex]+b1x[tex]^{3}[/tex]+c1x[tex]^{2}[/tex]+d1x+e1
+
c2x[tex]^{2}[/tex]+d2x+e2

which i think are the 2 even degree polynomials so i add them and see if the answer is also in P[tex]_{5}[/tex](x)
a1x[tex]^{4}[/tex]+b1x[tex]^{3}[/tex]+(c1+c2)x[tex]^{2}[/tex]+(d1+d2)x+(e1+e2)
since the resulting polynomial is still of degree 4 it is in P[tex]_{5}[/tex](x) and so is a subspace?

(ii)by much the same reasoning

b1x[tex]^{3}[/tex]+c1x[tex]^{2}[/tex]+d1x+e1 is of degree 3 and so is not a subspace?

i don't understand questions (iii) & (iv), please pointers anyone then i'll try to do them
 
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  • #2


Definition of a subspace: It's a vector space, and a subset of your original vector space. The common way to test that something is a subspace is to see if it's closed under linear transformations.

So for (ii), a degree 3 polynomial is always of degree less than 5, so you have to test if it's closed under linear combinations. Similarly for (i), you didn't add two arbitrary even degree polynomials, and that can get you in trouble. For example, what about these two:

x4+x-1 and -x4? When you add them are you still in the subset of even degree polynomials? Try something similar for (ii).

For (iii), I'm not sure what they want, but it looks like they're asking if {0} is a subspace maybe

For (iv), this is the set of all p(x) such that there exists some r a real number with p(r)=0 (as opposed to, say, x2+1 which has no real roots). So you need to see if adding two polynomials which each has a real root gives another polynomial which has a real root
 

Related to Subspaces in Polynomial P_5(x) of Degree < 5

1. What is a subspace in Polynomial P_5(x) of Degree < 5?

A subspace in Polynomial P_5(x) of Degree < 5 is a subset of the polynomial space that satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.

2. How is a subspace in Polynomial P_5(x) of Degree < 5 different from a polynomial of degree < 5?

A polynomial of degree < 5 is a single polynomial function, while a subspace in Polynomial P_5(x) of Degree < 5 is a collection of polynomial functions that satisfy the properties of a vector space. This means that a subspace can contain multiple polynomials of degree < 5.

3. How can you determine if a polynomial is in a subspace of Polynomial P_5(x) of Degree < 5?

A polynomial is in a subspace of Polynomial P_5(x) of Degree < 5 if it satisfies the properties of a vector space. This means that it must be closed under addition and scalar multiplication, and contain the zero vector. To determine this, you can perform operations on the polynomial and check if the resulting polynomial also satisfies these properties.

4. Can a subspace in Polynomial P_5(x) of Degree < 5 contain polynomials of different degrees?

No, a subspace in Polynomial P_5(x) of Degree < 5 can only contain polynomials of degree < 5. This is because the subspace must satisfy the properties of a vector space, and adding polynomials of different degrees would violate the closure under addition property.

5. What are some applications of subspaces in Polynomial P_5(x) of Degree < 5?

Subspaces in Polynomial P_5(x) of Degree < 5 have various applications in fields such as engineering, physics, and computer science. They can be used to model and solve problems involving polynomial functions, such as in digital signal processing, control systems, and data interpolation. They are also essential in understanding and analyzing algorithms for polynomial-time problems in computer science.

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