- #1

gtfitzpatrick

- 379

- 0

(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