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

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

Sagot :

GinnyAnsweridn
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]
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