Completely factor 𝑍 cubed plus six 𝑍 squared plus six plus 𝑍.
In order to factor, we need to put this in descending order, standard form. So the highest power, the highest exponent, will go first, then the next highest and the next highest, and so on. So 𝑍 cubed is first, then plus six 𝑍 squared plus 𝑍 plus six.
Now there are four terms here. So we can factor by grouping. So we will group the first two terms together and the last two terms together. And now we will take a greatest common factor out of each set of parentheses.
Out of 𝑍 cubed and six 𝑍 squared, we can take out a 𝑍 squared. And if we would divide each of these terms by 𝑍 squared — essentially take it out — we will be left with 𝑍 plus six. Now out of the next set of parentheses, we can only take out a one. So if we would divide each term by one, we will be left with 𝑍 plus six.
So now our GCFs will become one factor, so 𝑍 squared and positive one. And then the matching parentheses will be the other factor, which will be 𝑍 plus six. Therefore, after completely factoring, we get an answer of 𝑍 squared plus one times 𝑍 plus six.