Elementary Algebra!!!?

Well,

first of all, welcome on the forum from (maybe the only) the french guy in the zone !! ;-)

"elementary" means here : elements of algebra and not that it is this simple...

so :
B ={1,1+x,1-x^2,2+x^3} is a basis of P3.
polynomials of B are of degree 0, 1, 2 and 3
therefore : B is a generating family
let be :
u0 = 1,
u1 = 1+x,
u2 = 1-x^2,
u3 = 2+x^3
and v0=1, v1=x, v2=x^2, v3=x^3 the vectors of the "canonic" basis (C) of P3
then :
u0=v0, u1=v0+v1, u2=v0-v2, u3=2v0+v3
giving :
v0=u0, v1=u1-u0, v2=u0-u2, v3=u3 - 2u0
and we see that the correspondance between basis (B) and (C) is bijective
(can also be proven by calculating the determinant of the 4x4 transformation matrix)
therefore :
(B) is a basis

Let be S={ x+x^2 , x-x^2}
a) S is a linearly independent set:
let be a and b any two reals so that :
a(x+x^2) + b(x-x^2) = 0 valid for any x € R
then :
(a+b)x + (a-b)x^2 = 0 valid for any x € R
and :
the important rule is :
a polynomial p(x) = 0 if and only if all its coefficient are = 0
therefore we obtain :
a+b=0
a-b=0
adding : 2a = 0 ==> a = 0
substracting : 2b = 0 ==> b = 0
this yields that :
S is a linearly independent set

b) Extend S to a basis of P3 by adding suitable polynomials from the basis B.
vectors/polynomials of S have degrees 1 and 2
so, we need to add :
u0=1 of degree 0 (constant) and
u3=2+x^3 of degree 3
to build a basis

one question at a time

et voilà, mademoiselle !! ;-)

hope it' ll help !!

PS: if you want good answers, do not forget to give best Answers, if the answer deserves it.