# Math Help - Abstract Algebra

1. ## Abstract Algebra

How can I show that 1 is not a linear combination of polynomials 2, and x in Z[x].... It is a bit easier to think of it as showing that you can't find polynomials f, g in Z[x] where 2f(x) + xg(x) = 1..Thanks.. I don't think the division algorithm is helpful here.

2. Originally Posted by JohnStaphin
How can I show that 1 is not a linear combination of polynomials 2, and x in Z[x].... .
What is the constant term of the polynomial $2f(x) + xg(x)$?
Whatever it is, it must be an even number. But $1$ is odd.
Which is a contradiction.

3. That might be a good point, but who says there is a constant term...and even if there was...that can't be enough to show that the other terms don't add up to a negative odd number..which could cancel down to 1..

4. Originally Posted by JohnStaphin
That might be a good point, but who says there is a constant term...and even if there was...that can't be enough to show that the other terms don't add up to a negative odd number..which could cancel down to 1..
$f(x) = a_0 + a_1 x + ... + a_n x^n$
$g(x) = b_0 + b_1 x + ... + b_m x^m$

Then $2f(x) + xg(x) = 2a_0 + (2a_1 + b_0)x + (2a_2 + b_1)x^2 + ...$

But it is impossible for $2a_0 = 1$.

5. oh... >< You're right. Thanks