Which shows one way to determine the factors of [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] by grouping?

A. [tex]\(2x^2(6x - 1) + 3(6x - 1)\)[/tex]
B. [tex]\(2x^2(6x - 1) - 3(6x - 1)\)[/tex]
C. [tex]\(6x(2x^2 - 3) - 1(2x^2 - 3)\)[/tex]
D. [tex]\(6x(2x^2 + 3) + 1(2x^2 + 3)\)[/tex]



Answer :

To find the factors of the polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] by grouping, we can follow these steps:

1. Group the terms:
[tex]\[ 12x^3 - 2x^2 + 18x - 3 = (12x^3 + 18x) - (2x^2 + 3) \][/tex]

2. Factor out the greatest common factor (GCF) from each group:
- For the first group, [tex]\(12x^3 + 18x\)[/tex]:
[tex]\[ 12x^3 + 18x = 6x(2x^2 + 3) \][/tex]
- For the second group, [tex]\(-(2x^2 + 3)\)[/tex]:
[tex]\[ -(2x^2 + 3) = -1(2x^2 + 3) \][/tex]

3. Rewrite the expression with these factorizations:
[tex]\[ 12x^3 - 2x^2 + 18x - 3 = 6x(2x^2 + 3) - 1(2x^2 + 3) \][/tex]

4. Factor out the common binomial factor [tex]\((2x^2 + 3)\)[/tex]:
[tex]\[ 12x^3 - 2x^2 + 18x - 3 = (6x - 1)(2x^2 + 3) \][/tex]

Therefore, the polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] is factored as:
[tex]\[ 6x(2x^2 + 3) + 1(2x^2 + 3) \][/tex]

So, the correct choice is:
[tex]\[ \boxed{6x\left(2x^2+3\right)+1\left(2x^2+3\right)} \][/tex]