Answer :

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

1. Identify and Factor Out the Greatest Common Factor (GCF):

First, look for the greatest common factor in all the terms of the polynomial. In this case, each term ([tex]\(10x^3\)[/tex], [tex]\(-25x^2\)[/tex], and [tex]\(-15x\)[/tex]) has a common factor of [tex]\(5x\)[/tex].

Factor [tex]\(5x\)[/tex] out of each term:

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

2. Factor the Quadratic Expression:

Now, we need to factor the quadratic expression [tex]\(2x^2 - 5x - 3\)[/tex]. To do this, we look for two numbers that multiply to the product of the leading coefficient and the constant term ([tex]\(2 \times -3 = -6\)[/tex]) and add up to the middle coefficient ([tex]\(-5\)[/tex]).

The numbers [tex]\(-6\)[/tex] and [tex]\(1\)[/tex] work because:

- [tex]\(-6 \times 1 = -6\)[/tex]
- [tex]\(-6 + 1 = -5\)[/tex]

Re-write the middle term [tex]\(-5x\)[/tex] using [tex]\(-6x\)[/tex] and [tex]\(1x\)[/tex]:

[tex]\[
2x^2 - 5x - 3 = 2x^2 - 6x + x - 3
\][/tex]

Group the terms and factor by grouping:

[tex]\[
= (2x^2 - 6x) + (x - 3)
\][/tex]

Factor out the GCF from each group:

[tex]\[
= 2x(x - 3) + 1(x - 3)
\][/tex]

Since [tex]\((x - 3)\)[/tex] is common in both groups, factor it out:

[tex]\[
= (x - 3)(2x + 1)
\][/tex]

3. Combine the Factors:

Now, replace the quadratic part in the original expression:

[tex]\[
10x^3 - 25x^2 - 15x = 5x(2x^2 - 5x - 3) = 5x(x - 3)(2x + 1)
\][/tex]

Therefore, the polynomial [tex]\(10x^3 - 25x^2 - 15x\)[/tex] is factored as [tex]\(5x(x - 3)(2x + 1)\)[/tex].

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