showing an easy set is an ideal

Question

I'm having troubles understanding the definition of an ideal. I found this example online, but the author did not explain the steps and just stated it was an ideal. I hope someone could show me the steps

Let $g_1,...,g_n \in k[x]$ where $k[x]$ denotes the set of polynomials over a field $k$.

Let $J = \{ f_1g_1 +...+f_ng_n : f_j \in k[x] \}$, show that $J$ is an ideal:

I have shown:

1. Let $f_j = 0 $ then $0 \in J$

I must show:

2. If $f,g \in J$ then $f+g \in J$.

3. If $g \in J$ and $f \in k[x]$ then $fg \in J$

Any step by step approaches to showing 2 and 3 would be great thank you

Answer 1

$$\begin{align}&\sum_{i=1}^n f_ig_i+\sum_{i=1}^n q_ig_i=\sum_{i=1}^n(f_i+q_i)g_i\in J\\ &k\sum_{i=1}^n f_ig_i=\sum_{i=1}^n(kf_i)g_i\in J\end{align}$$