To factor the polynomial expression [tex]\(15x^3 - 5x^2 + 6x - 2\)[/tex] by grouping, follow these steps:
1. Identify and Group Terms:
Let's group the terms in pairs to make it easier to see any common factors:
[tex]\[
(15x^3 - 5x^2) + (6x - 2)
\][/tex]
2. Factor Out the Greatest Common Factor (GCF) from Each Group:
For the first group [tex]\((15x^3 - 5x^2)\)[/tex]:
[tex]\[
15x^3 - 5x^2 = 5x^2(3x - 1)
\][/tex]
For the second group [tex]\((6x - 2)\)[/tex]:
[tex]\[
6x - 2 = 2(3x - 1)
\][/tex]
Now, the expression looks like this:
[tex]\[
5x^2(3x - 1) + 2(3x - 1)
\][/tex]
3. Factor Out the Common Binomial Factor:
Notice that [tex]\((3x - 1)\)[/tex] is a common factor in both terms. We can factor [tex]\((3x - 1)\)[/tex] out:
[tex]\[
(3x - 1)(5x^2 + 2)
\][/tex]
4. Conclusion:
The factored form of the given polynomial [tex]\(15x^3 - 5x^2 + 6x - 2\)[/tex] is:
[tex]\[
(3x - 1)(5x^2 + 2)
\][/tex]
So, the correct answer is:
[tex]\(\left(5 x^2+2\right)(3 x-1)\)[/tex].
