Sure, let's find a way to determine the factors of the polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] by grouping.
### Step-by-Step Solution:
#### Step 1: Original Polynomial
We start with the given polynomial:
[tex]\[ 12x^3 - 2x^2 + 18x - 3 \][/tex]
#### Step 2: Separate into Two Groups
Group the polynomial into two parts:
[tex]\[ (12x^3 - 2x^2) + (18x - 3) \][/tex]
#### Step 3: Factor Each Group Separately
For the first group [tex]\( (12x^3 - 2x^2) \)[/tex]:
- Factor out the common term [tex]\(2x^2\)[/tex]:
[tex]\[ 2x^2(6x - 1) \][/tex]
For the second group [tex]\( (18x - 3) \)[/tex]:
- Factor out the common term [tex]\(3\)[/tex]:
[tex]\[ 3(6x - 1) \][/tex]
#### Step 4: Combine Both Groups with Common Factor
- Notice that both factored groups contain the common factor [tex]\((6x - 1)\)[/tex]:
[tex]\[ 2x^2(6x - 1) + 3(6x - 1) \][/tex]
#### Step 5: Verify the Possible Answers
- [tex]\(2 x^2(6 x-1)+3(6 x-1)\)[/tex]
- [tex]\(2 x^2(6 x-1)-3(6 x-1)\)[/tex]
- [tex]\(6 x (2 x^2-3)- 1(2 x^2-3)\)[/tex]
- [tex]\(6 x (2 x^2+3)+1(2 x^2+3)\)[/tex]
Based on our steps, the correct expression that shows how the factors are grouped for the polynomial [tex]\( 12x^3 - 2x^2 + 18x - 3 \)[/tex] is:
[tex]\[ 2 x^2(6 x - 1) + 3(6 x - 1) \][/tex]
Thus, the answer is:
[tex]\[ \boxed{1} \][/tex]
