College

Factor the four-term polynomial by grouping:

[tex]15x^3 - 25x^2 + 6x - 10[/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. [tex]15x^3 - 25x^2 + 6x - 10 = \square[/tex]

B. The polynomial is not factorable by grouping.

Answer :

To factor the polynomial [tex]\(15x^3 - 25x^2 + 6x - 10\)[/tex] by grouping, follow these steps:

1. Group the terms: Divide the polynomial into two groups:
- First group: [tex]\(15x^3 - 25x^2\)[/tex]
- Second group: [tex]\(6x - 10\)[/tex]

2. Factor the first group: [tex]\(15x^3 - 25x^2\)[/tex]
- Find the greatest common factor (GCF) of [tex]\(15x^3\)[/tex] and [tex]\(25x^2\)[/tex], which is [tex]\(5x^2\)[/tex].
- Factor out [tex]\(5x^2\)[/tex]: [tex]\(5x^2(3x - 5)\)[/tex]

3. Factor the second group: [tex]\(6x - 10\)[/tex]
- Find the greatest common factor (GCF) of [tex]\(6x\)[/tex] and [tex]\(10\)[/tex], which is [tex]\(2\)[/tex].
- Factor out [tex]\(2\)[/tex]: [tex]\(2(3x - 5)\)[/tex]

4. Reform the original expression: Now the expression looks like this:
[tex]\[
5x^2(3x - 5) + 2(3x - 5)
\][/tex]

5. Factor by grouping: Notice that the expression [tex]\(3x - 5\)[/tex] is common in both terms. Factor out the common factor [tex]\(3x - 5\)[/tex]:
[tex]\[
(5x^2 + 2)(3x - 5)
\][/tex]

Therefore, the polynomial [tex]\(15x^3 - 25x^2 + 6x - 10\)[/tex] can be factored by grouping as [tex]\((5x^2 + 2)(3x - 5)\)[/tex]. So the correct choice is:

A. [tex]\(15 x^3 - 25 x^2 + 6 x - 10 = (5x^2 + 2)(3x - 5)\)[/tex]

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