To factor the given expressions, we will identify common factors and use factoring techniques where applicable.
### 1. Factor [tex]\( 5x + 5y + 5z \)[/tex]
We observe that all terms have a common factor of [tex]\( 5 \)[/tex].
[tex]\[ 5x + 5y + 5z = 5(x + y + z) \][/tex]
### 2. Factor [tex]\( ax + ay + 9z \)[/tex]
In this case, there isn't a single common factor for all terms. Each pair of terms has a different common factor, but no single factor can be factored out from all three terms together. Hence, this expression is already in its simplest factored form.
[tex]\[ ax + ay + 9z \][/tex]
### 3. Factor [tex]\( 4x^3 + 8x^2 + 12x \)[/tex]
First, observe that each term can be divided by [tex]\( 4x \)[/tex].
[tex]\[ 4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3) \][/tex]
### 4. Factor [tex]\( 6x + 18y - 9z \)[/tex]
We identify that each term has a common factor of [tex]\( 3 \)[/tex].
[tex]\[ 6x + 18y - 9z = 3(2x + 6y - 3z) \][/tex]
### 5. Factor [tex]\( 3a + 6a^2 + 12 \)[/tex]
We recognize that each term has a common factor of [tex]\( 3 \)[/tex].
[tex]\[ 3a + 6a^2 + 12 = 3(2a^2 + a + 4) \][/tex]
Thus, the factored forms of the given expressions are:
[tex]\[
\begin{aligned}
&\text{1. } 5x + 5y + 5z &&= 5(x + y + z) \\
&\text{2. } ax + ay + 9z &&= ax + ay + 9z \\
&\text{3. } 4x^3 + 8x^2 + 12x &&= 4x(x^2 + 2x + 3) \\
&\text{4. } 6x + 18y - 9z &&= 3(2x + 6y - 3z) \\
&\text{5. } 3a + 6a^2 + 12 &&= 3(2a^2 + a + 4) \\
\end{aligned}
\][/tex]
