Prove or disprove: there exists a basis (p_0, p_1, p_2, p_3)of P_3 (F) such that

Question

Prove or disprove: there exists a basis (p_0, p_1, p_2, p_3)of P_3 (F) such that none of the polynomials p_0, p_1, p_2, p_3 hasdegree 2.
---- Is the following proof correct? ----
Let p_0, p_1, p_2, p_3 be elements of P_3(F) s.t.
p_o (x) = 1, p_1 (x) = x, p_2 (x) = x^2 + x^3, p_3(x) = x^3.
None of the polynomials are degree 2 although(p_0,p_1,p_2,p_3) is clearly spanning P_3 (F) with dimP_3(F) = 4and forms a basis. Hence proved. Prove or disprove: there exists a basis (p_0, p_1, p_2, p_3)of P_3 (F) such that none of the polynomials p_0, p_1, p_2, p_3 hasdegree 2.
---- Is the following proof correct? ----
Let p_0, p_1, p_2, p_3 be elements of P_3(F) s.t.
p_o (x) = 1, p_1 (x) = x, p_2 (x) = x^2 + x^3, p_3(x) = x^3.
None of the polynomials are degree 2 although(p_0,p_1,p_2,p_3) is clearly spanning P_3 (F) with dimP_3(F) = 4and forms a basis. Hence proved.

Explanation / Answer

SEE MY COMMENTS BELOW IN CAPITALS .
AT THE OUTSET THE QUESTION WAS NOT WELL FRAMED .
ON ONE HAND IT SAYS
there exists a basis (p_0, p_1, p_2, p_3) of P_3 (F) such thatone
of the polynomials p_0, p_1, p_2, p_3 has degree2.
SINCE THE SYMBOL P_2 ITSELF MEANS A POLYNOMIAL OFDEGREE 2 ..
THEN WHERE IS THE QUESTION IN SAYING THAT THE BASIS HAS A
POLYNOMIAL OF DEGREE 2??
IT SHOULD BE there exists a basis (X,Y,Z,S) of P_3 (F) such that
one of the polynomials X,Y,Z,S has degree2....
.........OR..............
A basis (X , Y, Z,S) of P_3 (F) SHALL BE such that one of thepolynomials X, Y, Z, S has degree 2....
OR SOME SUCH THING ....


Question Details: there exists a basis (p_0, p_1, p_2, p_3) of P_3 (F) such thatone
of the polynomials p_0, p_1, p_2, p_3 has degree2.
SINCE THE SYMBOL P_2 ITSELF MEANS A POLYNOMIAL OFDEGREE 2 ..
THEN WHERE IS THE QUESTION IN SAYING THAT THE BASIS HAS A
POLYNOMIAL OF DEGREE 2??
IT SHOULD BE there exists a basis (X,Y,Z,S) of P_3 (F) such that
one of the polynomials X,Y,Z,S has degree2....
.........OR..............
A basis (X , Y, Z,S) of P_3 (F) SHALL BE such that one of thepolynomials X, Y, Z, S has degree 2....
OR SOME SUCH THING ....

there exists a basis (X,Y,Z,S) of P_3 (F) such that
one of the polynomials X,Y,Z,S has degree2....
.........OR..............
A basis (X , Y, Z,S) of P_3 (F) SHALL BE such that one of thepolynomials X, Y, Z, S has degree 2....
OR SOME SUCH THING ....


Prove or disprove: there exists a basis (p_0, p_1, p_2, p_3)of P_3 (F) such that one of the polynomials p_0, p_1, p_2, p_3 hasdegree 2.
---- Is the following proof correct?
YOU BETTER STATE FIRST ....THE STATEMENT IS TRUE OR FALSE
HERE WHAT DO YOU WANT TO SAY ?
YOU GAVE A COUNTEREXAMPLE BELOW TO SHOW THAT A BASIS FOR P3NEED NOT CONTAIN A P2 POLYNOMIAL.THE EXAMPLE IS PERFECTLYCORRECT.
BUT IS THIS COUNTER EXAMPLE RELEVANT ??
THE QUESTION OR PROPOSITION IS THAT " there EXISTS a basis (p_0, p_1, p_2, p_3) of P_3 (F)such that one of the polynomials p_0, p_1, p_2, p_3
has degree 2."
THE STATEMENT IS TRUE..
SINCE THE STANDARD BASIS OF P3 NAMELY [1,X,X2,X3] IS ONE WHICH HAS
A P2 POLYNOMIAL IN IT ..PROVING THE EXISTENCE OF A P2 POLYNOMIAL INA
BASIS FOR P3.
SO FIRST MAKE UP YOUR MIND WHETHER THE GIVEN STATEMENT IS TRUE ORFALSE
AND THEN GIVE A PROOF IF IT IS CORRECT AND GIVE A COUNTEREXAMPLE ORPROOF
IF IT IS FALSE.
there EXISTS a basis (p_0, p_1, p_2, p_3) of P_3 (F)such that one of the polynomials p_0, p_1, p_2, p_3
has degree 2."
THE STATEMENT IS TRUE..
SINCE THE STANDARD BASIS OF P3 NAMELY [1,X,X2,X3] IS ONE WHICH HAS
A P2 POLYNOMIAL IN IT ..PROVING THE EXISTENCE OF A P2 POLYNOMIAL INA
BASIS FOR P3.
SO FIRST MAKE UP YOUR MIND WHETHER THE GIVEN STATEMENT IS TRUE ORFALSE
AND THEN GIVE A PROOF IF IT IS CORRECT AND GIVE A COUNTEREXAMPLE ORPROOF
IF IT IS FALSE.
----
Let p_0, p_1, p_2, p_3 be elements of P_3(F) s.t.
p_o (x) = 1, p_1 (x) = x, p_2 (x) = x^2 + x^3, p_3(x) = x^3.
None of the polynomials are degree 2 although(p_0,p_1,p_2,p_3) is clearly spanning P_3 (F) with dimP_3(F) = 4and forms a basis.
THIS EXAMPLE AS I SAID BEFORE IS PERFECTLY CORRECT BY IT SELF. BUTNOT RELEVANY IN THE PRESENT CASE .
THIS WOULD HAVE BEEN RELEVANT IF THE QUESTION WAS TO PROVE ORDISPROVE THAT
1. A basis (X , Y, Z,S) of P_3 (F) SHALL BE such thatone of the polynomials X, Y, Z, S has degree 2.



Hence proved. 1. A basis (X , Y, Z,S) of P_3 (F) SHALL BE such thatone of the polynomials X, Y, Z, S has degree 2.

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