If R is a commutative ring with unity

To show that $R[x]$ is a commutative ring with unity 1, we need to prove that it satisfies the following properties:
1. Addition and multiplication are well-defined.
2. $R[x]$ is closed under addition and multiplication.
3. Addition and multiplication are commutative.
4. Addition is associative and has an identity element.
5. Multiplication is associative and has an identity element.
6. Multiplication is distributive over addition.
Let's start with property 1. We define addition and multiplication in $R[x]$ as follows:
$(a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0)+(b_n{x}^n+b_{n-1}{x}^{n-1}+\cdots +b_{1}x+b_0)$
$=(a_n+b_n)x^n+(a_{n-1}+b_{n-1})x^{n-1}+\cdots +(a_1+b_1)x+(a_0+b_0)$
$(a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0)·(b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_{1}x+b_0)$
$=(a_n{b}_m)x^{n+m}+(a_n{b}_{m-1}+a_{n-1}{b}_m)x^{n+m-1}+\cdots +(a_1{b}_0+a_0{b}_1)x+a_0{b}_0$
We can easily check that these operations are well-defined and that $R[x]$ is closed under them. Now, let's check the remaining properties:
2. Closure under addition and multiplication: This follows directly from the definition of addition and multiplication in $R[x]$.
3. Commutativity of addition and multiplication: This also follows from the definition of addition and multiplication in $R[x]$. We can easily see that adding or multiplying two polynomials does not depend on the order of the terms.
4. Associativity and identity of addition: This follows from the fact that $R$ is a commutative ring with unity 1. We can easily check that the polynomial $0_R=0+0x+0x^2+\cdots$ serves as the additive identity in $R[x]$. Also, for any polynomial $f(x)\in R[x]$, we have $f(x)+0_R=f(x)$, which shows that $0_R$ is indeed an additive identity.
5. Associativity and identity of multiplication: This also follows from the fact that $R$ is a commutative ring with unity 1. The polynomial $1_R=1+0x+0x^2+\cdots$ serves as the multiplicative identity in $R[x]$. For any polynomial $f(x)\in R[x]$, we have $f(x)·1_R=f(x)$, which shows that $1_R$ is indeed a multiplicative identity.
6. Distributivity of multiplication over addition: This follows directly from the distributivity of multiplication over addition in $R$.
Therefore, $R[x]$ is indeed a commutative ring with unity 1.