Rings and Integral Domains

**Aryth** · September 16th 2008, 09:13 AM

Let $\mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\mathcal{Z}$ and the usual operations of addition and multiplication.

Let $S$ be the set of polynomials in $\mathcal{Z}[x]$ with roots at $x=0$ and $x=1$ . Prove that $S$ is a subring of $\mathcal{Z}[x]$ .

**ThePerfectHacker** · September 16th 2008, 09:37 AM

Originally Posted by Aryth

Let $\mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\mathcal{Z}$ and the usual operations of addition and multiplication.

Let $S$ be the set of polynomials in $\mathcal{Z}[x]$ with roots at $x=0$ and $x=1$ . Prove that $S$ is a subring of $\mathcal{Z}[x]$ .

Do it by definition.
For example, for additive closure say $f(x),g(x) \in S$ .
Then $f(x)+g(x) \in S$ since $f(0)+g(0) = 0 + 0 = 0$ and $f(1)+g(1) = 0 + 0 = 0$ .

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