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Factor [tex][tex]$15x^3 - 5x^2 + 6x - 2$[/tex][/tex] by grouping. What is the resulting expression?

A. [tex]\left(5x^2 + 2\right)(3x - 1)[/tex]
B. [tex]\left(5x^2 - 2\right)(3x + 1)[/tex]
C. [tex]\left(15x^2 + 2\right)(x - 1)[/tex]
D. [tex]\left(15x^2 - 2\right)(x + 1)[/tex]

Sagot :

GinnyAnswer
To factor the polynomial [tex]\(15x^3 - 5x^2 + 6x - 2\)[/tex] by grouping, we can proceed with the following steps:

1. Group the polynomial into two pairs:
[tex]\[ (15x^3 - 5x^2) + (6x - 2) \][/tex]

2. Factor out the greatest common factor (GCF) from each pair:
- For the first group, [tex]\(15x^3 - 5x^2\)[/tex], the GCF is [tex]\(5x^2\)[/tex]:
[tex]\[ 15x^3 - 5x^2 = 5x^2(3x - 1) \][/tex]
- For the second group, [tex]\(6x - 2\)[/tex], the GCF is 2:
[tex]\[ 6x - 2 = 2(3x - 1) \][/tex]

3. Rewrite the polynomial using these common factors:
[tex]\[ 15x^3 - 5x^2 + 6x - 2 = 5x^2(3x - 1) + 2(3x - 1) \][/tex]

4. Notice that [tex]\((3x - 1)\)[/tex] is a common factor in both terms:
[tex]\[ = (3x - 1)(5x^2 + 2) \][/tex]

Therefore, the factored form of the polynomial [tex]\(15x^3 - 5x^2 + 6x - 2\)[/tex] is:
[tex]\[ (3x - 1)(5x^2 + 2) \][/tex]

Thus, the correct choice is:
[tex]\[ \left(5x^2 + 2\right)(3x - 1) \][/tex]
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