To determine the factors of the polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] by grouping, follow these steps:
1. Group the terms in pairs: Start by grouping the polynomial into two pairs of terms:
[tex]\[
(12x^3 - 2x^2) + (18x - 3)
\][/tex]
2. Factor each group: Next, factor out the greatest common factor (GCD) from each of the two groups.
- For the first group [tex]\(12x^3 - 2x^2\)[/tex]:
[tex]\[
12x^3 - 2x^2 = 2x^2(6x - 1)
\][/tex]
- For the second group [tex]\(18x - 3\)[/tex]:
[tex]\[
18x - 3 = 3(6x - 1)
\][/tex]
3. Combine the factored groups: Now, the polynomial can be written as:
[tex]\[
2x^2(6x - 1) + 3(6x - 1)
\][/tex]
4. Factor out the common binomial factor: Notice that [tex]\((6x - 1)\)[/tex] is a common factor in both terms:
[tex]\[
2x^2(6x - 1) + 3(6x - 1) = (6x - 1)(2x^2 + 3)
\][/tex]
By following these steps, the polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] is factored correctly. Therefore, the option that shows the grouping process correctly is:
[tex]\[
2x^2(6x - 1) + 3(6x - 1)
\][/tex]
Thus, the correct option is:
[tex]\[
2x^2(6x - 1) + 3(6x - 1)
\][/tex]
